Abstract

Even though it is well known that both the magnitude and detailed angular shape of scattering (phase function, PF), particularly in the backward angles, affect the color of the ocean, the current remote-sensing reflectance (Rrs) models typically account for the effect of its magnitude only through the backscattering coefficient (bb). Using 116 volume scattering function (VSF) measurements previously collected in three coastal waters around the U.S. and in the water of the North Atlantic Ocean, we re-examined the effect of particle PF on Rrs in four scenarios. In each scenario, the magnitude of particle backscattering (i.e., bbp) is known, but the knowledge on the angular shape of particle backscattering is assumed to increase from knowing nothing about the shape of particle PFs to partially knowing the particle backscattering ratio (Bp), the exact backscattering shape as defined by β˜p(γ90°) (particle VSF normalized by the particle total scattering coefficient), and the exact backscattering shape as defined by the χp factor (particle VSF normalized by the particle backscattering coefficient). At sun zenith angle=30°, the nadir-viewed Rrs would vary up to 65%, 35%, 20%, and 10%, respectively, as the constraints on the shape of particle backscattering become increasingly stringent from scenarios 1 to 4. In all four scenarios, the Rrs variations increase with both viewing and sun angles and are most prominent in the direction opposite the sun. Our results show a greater impact of the measured particle PFs on Rrs than previously found, mainly because our VSF data show a much greater variability in Bp, β˜p(γ90°), and χp than previously known. Among the uncertainties in Rrs due to the particle PFs, about 97% can be explained by χp, 90% by β˜p(γ90°), and 27% by Bp. The results indicate that the uncertainty in ocean color remote sensing can be significantly constrained by accounting for χp of the VSFs.

© 2017 Optical Society of America

1. INTRODUCTION

The study of ocean color is carried out by measuring the spectral radiance (Lw; Wm2sr1) leaving the ocean and normalizing it by the incident irradiance (Ed; Wm2) at the surface, forming the spectral remote-sensing reflectance (Rrs; sr1):

Rrs(λ,θs,θv,φ)=Lw(λ,θs,θv,φ)Ed(λ,θs),
where λ is the wavelength; θs and θv are the sun zenith angle and the sensor viewing angle, respectively; and ϕ is the azimuth angle of the sensor relative to the plane of sun. Through radiative transfer simulations, an approximate relationship was derived relating Rrs to two inherent optical properties (IOPs) of sea water, i.e., the absorption (a;m1) and backscattering (bb;m1) coefficients [15]:
Rrs=fQbba+bb.
Here, the f/Q factor describes the bidirectional reflectance distribution function, which depends not only on the sun-sensor geometry but also on the IOPs [6,7]. Though not explicitly contained in Eq. (2), the volume scattering function (VSF or β; m1sr1) plays a fundamental role in ocean color [8,9].

The VSFs influence the remote sensing reflectance [9,10] through both the backscattering coefficient and the bidirectional reflectance distribution function. bb describes the overall magnitude of VSFs in the backward directions and is generally responsible for reflecting incident sunlight out of the water. Assuming azimuthal symmetry of scattering, bb can be calculated as

bb=2ππ/2πβ(γ)sinγdγ,
where γ is the scattering angle. The shape of VSFs influences the distribution of the underwater light field, which, in turn, impacts the Rrs. However, due to limited knowledge on the VSFs of natural particle assemblage, most of the current ocean color algorithms [6,7,11] have ignored the shape effect. In other words, for two waters with the same a and sun-sensor geometry, Rrs is assumed to be the same if the backscattering magnitudes are also the same, regardless of how the actual angular scattering might differ from each other. Optical closure studies, however, have shown that Rrs simulated using measured a and bb but with an assumed angular scattering shape could differ significantly from the measured Rrs [1217].

The shape of a VSF is typically described using the phase function (PF),

β˜(γ)=β(γ)b,
where b=2π0πβ(γ)sinγdγ, representing the total scattering coefficient. Because of the importance of backward scattering in ocean color [4,8], the shape of a VSF in the backward angles is often described using two additional parameters, the backscattering probability (fraction or ratio),
B=bbb=2ππ/2πβ˜(γ)sinγdγ,
and the χ(γ) factor [1821],
χ(γ90°)=bb2πβ(γ)=B2πβ˜(γ).
While B can be considered as a measure of the general backward shape of a VSF, both the χ factor and β˜(γ90°) describe its exact backward shape. From an ocean color perspective, the χ factor is more relevant because it is normalized by the backscattering coefficient. Note that both B and the χ factor can be derived directly from β˜.

The angular shape of β˜ is contributed by both molecules and particles. Since the scattering by pure seawater is relatively well known [22,23], our focus in this study is on the effect of angular scattering by the particles, which represents the contribution by everything other than pure water and sea salt molecules and will be denoted by a subscript p. While Gordon [24] found that the PFs at angles <15° have a <5% impact on Rrs, other studies have shown that the VSF shapes at larger angles could have a significant impact.

Bulgarelli et al. [12] found 40% variation among simulated upwelling radiance using the Fournier–Forand (FF) [2527] PFs with Bp=0.011 and 0.033 and the average Petzold PF, which was estimated using the measurements taken by Petzold [28] in the San Diego harbor and has a Bp=0.018. Similarly, Tzortziou et al. [17] found Lw simulated using the average Petzold PF can differ by 50% from Lw measured in a water with considerably smaller Bp values, with an average=0.0128±0.0032. Using FF and a few other theoretical PFs that have the same Bp value as the average Petzold PF, Mobley et al. [25] found that simulated Rrs vary only by 10%. They further compared Rrs among measured, simulated with measured VSFs, and simulated with Bp-equal FF PF, and concluded that “… the exact shape of the phase function in backscatter directions does not greatly affect the light field, so long as … the phase function have [sic] the correct backscatter fraction.” The conclusion has since been tested. Chami et al. [29] found up to 20% differences in Rrs simulated using their measured PFs and the corresponding Bp-equal FF PFs. They also showed that an increase of the VSF between 10° and 100° could lead to increased reflectance due to multiple scattering. Tonizzo et al. [16] also compared Rrs simulated using measured PFs and the corresponding Bp-equal FF PFs over a range of case I and II waters and found an average difference of 20%.

The phase functions used in these studies, some of which are based on simplified theoretical models, are limited in their representation of natural variability of the VSFs. In addition, these studies considered only the general backscattering fraction, not the full VSF shapes, while theoretical derivation [8,9] indicates that the influence on Rrs arises from both the magnitude and the exact shape of the VSF in the backward directions. For example, recent studies by Pitarch et al. [15] and Lefering et al. [14] showed that accounting for the detailed shape of PFs is important in achieving optical closure in Rrs.

We have measured VSFs over various coastal waters of the U.S. and in clear waters of the North Atlantic Ocean [30,31], with chlorophyll concentration ranging from 0.07 to 40μg/L. The data show that the natural variability of particle VSFs is greater than what had been known or assumed previously in terms of both the backscattering probability and the detailed backward shape (Fig. 1). For example, Bp values estimated from our measured VSFs vary over a factor of 30 from 0.0015 to 0.0543, which is much greater than the range of 0.0073–0.0194 estimated from Petzold’s measurements and the range of 0.002–0.020 measured in Tonizzo et al. [16], but comparable to the range of 0.0015–0.0454 reported in Mankovsky and Haltrin [34,35] over the oceans and lakes worldwide.

 figure: Fig. 1.

Fig. 1. (a) Comparison of particle PFs derived from: our measurements (gray lines); samples of historical Petzold’s measurements in clear water of the Tongue of the Ocean (black dotted line), in California offshore coastal water (black dashed line), and in very turbid water of San Diego harbor (black dotted-dashed line) [32]; the average particle PF of Petzold’s measurements in San Diego harbor [33] (black solid line); the analytical Fournier–Forand (FF) formula having a Bp=1.83% [25] (blue dashed line); and a series of particle PFs generated by linear mixing of Bp=1% FF PF and the average Petzold PF, as used in Lee et al. [11] (green solid lines). The red line is one of the measured particle PFs referred to in Fig. 5. To highlight the variations, the x axes are in logarithmic scale for angles <30° and in linear scales for larger angles. (b) Corresponding χp factors. (c) Histogram of Bp calculated from measured particle PFs.

Download Full Size | PPT Slide | PDF

Therefore, we believe that the impact of angular scattering by oceanic particles on Rrs may have been underestimated. Using the latest measurements of the VSFs in a wide range of natural environments, we have re-examined the uncertainty that the particle PFs might have on the remote sensing of the color of the ocean. Our study differs from previous ones in two aspects. First, we are interested in the natural variability of particle PFs and their impact on Rrs. Therefore, we used only PFs derived from the measurements. Second, we examined the impact on Rrs by both general backward shape as defined by Bp and the exact backward shape as well. Mobley et al. [25] and Pitarch et al. [15] also looked into the exact shape of PFs, but they used only a few analytical PFs. While they are easy to generate, FF PFs with varying Bp values [25] or other analytical PF models may not cover full variability of VSF shapes of natural particles. And some of them may not even be suitable to represent the angular scattering by oceanic particles [15].

2. METHODOLOGY

A. Measured VSFs

The VSFs at or near the surface (within 1 optical depth) were measured in the field using a prototype multispectral volume scattering meter (MVSM) and a LISST-100X sensor (Sequoia Scientific Inc., Bellevue, Washington DC). The details of these measurements have been reported [30,31]. The near full angular VSFs were generated by combining the LISST-100X data at 30 near-forward angles (0.07°–9.45°) and the MVSM data at 0.25° angular resolution from 9.5° to 179°, both at 532 nm [30]. Laboratory experiments using polystyrene spheres showed the combined LISST and MVSM has an overall uncertainty of 5% in resolving the VSFs over the entire angular range [30] and inter-instrument comparison shows differences <10% between the combined VSFs and the scattering measurements by an ac-9 or ac-s (WET Labs Inc., Philomath, Oregon), an ECO-VSF (WET Labs Inc.), and a HydroScat-6 (HOBI Labs Inc., Bellevue, Washington DC) [31,36]. For this study, we used a total of 116 VSFs collected in three coastal waters of the United States (Chesapeake Bay in 2009, Mobile Bay in 2009, and Monterey Bay in 2010) and in the North Atlantic Ocean during NASA’s Ship Aircraft Bio-Optical Research (SABOR) cruise in 2014. The VSFs due to seawater were calculated following Zhang et al. [22] using concurrently measured temperature and salinity and subtracted from the bulk VSFs, generating βp’s, which were used in the following analysis.

To be used in HydroLight [37], a measured βp needs to be interpolated logarithmically between 0.1° and 180° at every 0.1°. In addition, the βp at near-forward angles need to be adjusted to ensure that their values at angles <0.1° follow a power-law function:

βp(γ)=Aγm,
where A and m are estimated using βp values at γ1=0.1° and γ2=0.2°.

The particle scattering coefficient is then computed as

bp=2π2mβp(γ1)γ12+2π0.1πβp(γ)sinγdγ.

A precondition, m<2, is necessary for Eq. (8) to be physically valid [37]. There are 16 (out of 116) measured VSFs with m2. For these VSFs, we modified βp(γ1) to ensure m=1.9. The tests were performed with those VSFs with m2 by changing m values between 1 to 1.9, and the results showed that the differences on HydroLight-simulated Rrs were <1.8%. These modified βp were used in generating the PF shapes in Fig. 1 and in the rest of this study.

B. HydroLight Simulation

We used HydroLight 5.2 to simulate Rrs following the methods described in the IOCCG report No. 5 [38]. Briefly, the absorption coefficients and backscattering coefficients of particles (ap and bbp) were generated as a function of chlorophyll concentration in 20 discrete values between 0.03 and 30.0μg/L, and at each value, 25 sets of spectral particle absorption and backscattering coefficients (ap and bbp) at 532 nm were generated, providing a total of 500 sets of ap and bbp. The values of bb/(a+bb) at 532 nm are between 0.03 and 0.34. For each generated synthetic data set of ap and bbp, Rrs was simulated ingesting each of the 116 measured β˜p. Also, the particle scattering coefficients (bp) were estimated from bbp using Bp associated with each β˜p. Water is assumed homogeneous and infinitely deep, illuminated by a semi-empirical sky model (based on RADTRAN-X) with 0 cloud coverage and annual average climatology condition. The simulations were performed for θs ranging from 0° to 75° with an increment of 15°, θv ranging from 0° to 70° with an increment of 10°, and ϕ ranging from 0° to 180° with an increment of 15°. In total, we had 32,016,000 simulated Rrs over a range of optical and viewing conditions that we believe cover a sufficiently extensive variability that an ocean color sensor may encounter.

3. RESULTS

A. Rrs(θs=30°,θv=0°)

The Rrs simulated with different particle PFs at θs=30° and θv=0° are shown in Fig. 2. As expected, Rrs increases with bb/(a+bb). For a set of a and bb values, the variability of Rrs also increases with bb/(a+bb). This confirms that the impact of the particle VSF shapes on Rrs increases with bbp. In the following, we will examine the impact of the VSF shape in four scenarios with increasing constraint on the backscattering: (1) no knowledge of β˜p, (2) some knowledge of backscattering ratio, i.e., Bp, (3) some knowledge of exact backward shape of β˜p, i.e., β˜p(γ90°), and (4) some knowledge of both Bp and β˜p(γ90°). In all four scenarios, the particle backscattering coefficient bbp is assumed to be known. To quantify uncertainty for Rrs due to β˜p, we use pairwise percentage difference, dRrs estimated as

dRrs(a,bb,θs,θv,φ)=|Rrs(i,a,bb,θs,θv,φ)Rrs(j,a,bb,θs,θv,φ)|12[Rrs(i,a,bb,θs,θv,φ)+Rrs(j,a,bb,θs,θv,φ)]×100%,
where i, j denote two arbitrary phase functions. Given its definition in Eq. (9), dRrs describes the variation arising entirely from the changes in particle PF for a given set of a and bb in a given sun-viewing geometry.

 figure: Fig. 2.

Fig. 2. Rrs(θs=30°,θv=0°) simulated with measured particle phase functions (color coded by the measurement locations) are shown as a function of bb/(a+bb).

Download Full Size | PPT Slide | PDF

1. No Knowledge of β˜p

For two waters with the same bbp (and ap), the current ocean color models, e.g., Eq. (2), would predict the same Rrs for a given viewing geometry. Therefore, any variations shown in the simulated Rrs would represent the uncertainty caused by the difference in β˜p shapes between the two waters [Fig. 3(a)]. With the natural variability exhibited by the β˜p shown in Fig. 1, the median differences in Rrs are <15% but the maximum differences could reach >60%.

 figure: Fig. 3.

Fig. 3. Percentage difference between Rrs(θs=30°,θv=0°) simulated with two different particle phase functions. The whiskers boxes, showing the minimum, lower quartile, median, upper quartile, and maximum values, summarize the differences at the selected bb/(a+bb) values. (a) Simulations performed with same bbp values. (b) Simulations performed with same bbp values and phase functions having the same Bp values. The dotted line at 10% represents the maximum difference found in Ref. [25], the dashed line at 20% represents the maximum difference found in Ref. [29], and the dotted-dashed line at 40% represents the maximum difference found in Ref. [16]. (c) Simulations performed with same bbp values and very similar β˜p(γ90°) (logarithmic value within 1%). (d) Simulations performed with same bbp values and very similar χp factor (within 1%).

Download Full Size | PPT Slide | PDF

2. Knowledge of Bp

Among the 116 measured β˜p, we found 44 pairs that have the same Bp values (within 0.5%). Since bbp and a are assumed to be the same, the same Bp effectively means the same b (and c). With this additional constraint on the general backward shape, the median difference in Rrs decreased to <10%, and the maximum difference decreased to 35% [Fig. 3(b)]. Under the same set of constraints, Mobley et al. [13] found a difference of up to 10% [dotted line in Fig. 3(b)] using Petzold’s average β˜p and a few other theoretical PFs with the same Bp as the Petzold’s average β˜p. A difference of up to 20% [dashed line in Fig. 3(b)] was found by Chami et al. [29] using several measured PFs and the Bp-matched FF functions, while a similar study by Tonizzo et al. [16] found differences up to 40% [dot-dashed line in Fig. 3(b)]. Our results are consistent with these earlier studies, showing that the knowledge of Bp is not sufficient to explain all the variations in Rrs caused by PF shapes.

3. Knowledge of β˜p(γ90°)

Among the 116 measured β˜p, 66 pairs of measured particle PFs [Fig. 3(c)] have similar backward shape, with <1% difference between the logarithmic values of β˜p(γ90°). With this knowledge, the median difference in Rrs was lowered to <6% and the maximum difference to 20%.

4. Knowledge of Both Bp and β˜p(γ90°)

From Eq. (6), the same Bp and β˜p(γ90°) are equivalent to the same χp. Therefore, this scenario also represents the constraint of “knowledge of χp. Among the 116 measured β˜p, we found 25 pairs that have the χp factors similar to each other within 1% at all angles between 90° and 180°. The Rrs simulated with the same bbp and similar χp factors are very similar to each other, with a median difference <2% and a maximum difference of 10% [Fig. 3(d)].

Common to all four scenarios is that dRrs increases with bb/(a+bb), seemingly reaching a saturation at approximately bb/(a+bb)>0.2. As bb/(a+bb) increases, the backscattering will shift from water molecule-dominated to particle-dominated scattering, and therefore the shape of particle PFs will have an increasing impact on Rrs. However, when bb/(a+bb) is high enough, the multiple scattering will be increased so much that the impact on Rrs due to the shape differences between particle PFs will not increase anymore [29].

B. Rrs(θs,θv)

To examine how the impact of the particle PFs on Rrs varies with sun-viewing geometry, we plot the maximum percentage difference in Rrs in Fig. 4 as a function of sun zenith angles, viewing zenith, and azimuth angles. Without any knowledge on the shape of particle PF [corresponding to Fig. 3(a)], the maximum differences in Rrs vary between 52% and 107% [Fig. 4(a)]. With a knowledge of Bp [corresponding to Fig. 3(b)], the simulated Rrs show a maximum difference varying between 26% and 54% [Fig. 4(b)]. If the backward shape of β˜p is similar within 1% [corresponding to Fig. 3(c)], the maximum differences in Rrs vary between 17% and 30% [Fig. 4(c)]. A knowledge of both Bp and β˜p(γ90°) [corresponding to Fig. 3(d)] will reduce the maximum variations in Rrs between 8% and 16% [Fig. 4(d)]. For all four cases, the difference in Rrs follows the same geometric pattern: increasing with both the sun and viewing zenith angles, and being more prominent when viewing direction is opposite to the sun.

 figure: Fig. 4.

Fig. 4. Distribution of the maximum value of dRrs. In the polar plots, ao-56-24-6881-i001 indicates θs, radial direction represents θv, and polar direction represents ϕ. Column (a) results with same bbp; column (b) results with same bbp and Bp; column (c) results with same bbp and similar β˜p(γ90°) (logarithmic value within 1%); column (d) results with same bbp and similar χp factor (within 1%).

Download Full Size | PPT Slide | PDF

4. DISCUSSION AND CONCLUSIONS

Even though the commonly used Rrs models do not explicitly account for the shape of the VSFs [e.g., Eq. (2)], its impact on the color of the ocean is well recognized [4,8,23,39,40]. In fact, Jerlov [40] pointed out that the observed variation of the reflectance with sun angles is a direct consequence of the shape of the VSF. In Eqs. (4)–(6), three commonly used parameters describing the backward shape of a VSF are introduced: B, β˜(γ90°) and the χ factor. So, a logical question to ask is which of these three shape factors is more important in affecting Rrs.

Under quasi-single-scattering approximation (QSS), Gordon [41] showed the remote sensing reflectance (the detailed derivation is given by Mobley et al. [42]):

Rrs(θ,ϕ,θs)=1cosθv+cosθsβ(γs)a+bb,
where γs represents the single scattering angle formed between the viewing and sun vectors. Since γs is typically >90°, it is clear from Eq. (10) that Rrs is directly proportional to the backward VSF. Inserting the first equality of Eq. (6) into Eq. (10), we have
Rrs(θ,ϕ,θs)=12πχ(γs)1cosθv+cosθsbba+bb.
It is clear from Eq. (11) that when bb, and hence bbp, is constrained, Rrs is directly proportional to 1/χ(γs), assuming a is known in a fixed sun-viewing geometry. Since the χ factor for pure seawater is known [21], the uncertainty in χp is directly reflected in Rrs. Our result showed that a complete lack of knowledge of particle backscattering shape would induce approximately 15% median and >60% maximum differences in nadir-viewed Rrs [Fig. 3(a)]. On the other hand, Eq. (11) shows that if the χp factor is constrained, Rrs will be fixed. While this applies only to the QSS approximation, it does explain why the simulated Rrs are very similar to each other, with median difference <2% and maximum difference <10%, when the χp factor is constrained [Fig. 3(d)].

Inserting the second equality of Eq. (6) into Eq. (11), we have

Rrs(θ,ϕ,θs)=β˜(γs)B1cosθv+cosθsbba+bb.
Apparently, when only one of B and β˜ is constrained, there is still residual variance in Rrs that arises from the other factor that is not constrained. For example, our results show that the median and maximum values of this residual variance due to unknown β˜p are approximately 10% and 35%, respectively [Fig. 3(b)], and are 5% and 20%, respectively, if Bp is unknown [Fig. 3(c)].

As shown in Eq. (11), when either θv or θs increases, the value of 1cosθv+cosθs increases, and hence the impact of backward VSF increases. Therefore, the uncertainty due to the shape of VSF increases with both viewing and sun angles (Fig. 4).

For Bp, β˜p(γ90°), or χp, the field measurements indicate that the natural variability of the particle PFs is much greater than what has been assumed (Fig. 1). Consequently, the impact of the shape of the particle PFs on Rrs may have been underestimated. For example, Fig. 5 compares the Rrs predicted using the Lee’s et al. [11] model, which was developed based on a limited range of phase functions (green lines in Fig. 1), with the Rrs simulated using one particular phase function (red line in Fig. 1). Both Lee et al. [11] and our study followed the IOCCG Report No. 5 in generating the IOPs to drive the HydroLight. Since the final formula of the Lee et al. [11] model accounts for only a statistic average of the PFs used in their study, the scatter of the comparison shown in Fig. 5 is expected. However, the Lee et al. [11] model consistently over-predicted the Rrs simulated for this particular PF by approximately 30%–50%. While the comparison shown in Fig. 5 represents an extreme case with the differences close to the maximum shown in Fig. 3(a), it does illustrate the potential impact of the VSF shape on the color of the ocean.

 figure: Fig. 5.

Fig. 5. Comparison between Rrs simulated with one of the measured phase functions (red line in Fig. 1) by HydroLight and Rrs estimated using the Lee’s et al. [11] model at all viewing geometries (θs=075°, θv=070°, and ϕ=0180°).

Download Full Size | PPT Slide | PDF

Our results have demonstrated the increasing importance of Bp, β˜p(γ90°), and χp in regulating Rrs, and it will be of interest to quantify their respective importance in terms of fraction of total uncertainty in Rrs due to the VSF shapes that they can explain. To do this, we started with β˜p, which describes the general shape of a VSF, includes β˜p(γ90°), and can be used to derive both Bp and χp [Eqs. (5) and (6)]. Let σs denote the total uncertainty in Rrs explained by β˜p. In the first test with no constraint on the shape, Rrs were found to differ up to 65% for nadir-viewed geometries [Fig. 3(a)]. In other words, the maximum uncertainty due to VSF shapes is 65%, i.e., σs=0.65. A constraint on Bp was introduced in the second test, and, removing the uncertainty due to Bp, the maximum uncertainty due to VSF shapes was reduced to 35% [Fig. 3(b)], i.e., σs2σBp2=0.352, where σBp denotes the uncertainty in Rrs explained by Bp. A constraint on β˜p(γ90°) was introduced in the third test, and, removing the uncertainty due to β˜p(γ90°), the maximum uncertainty due to VSF shapes was reduced to 20% [Fig. 3(c)], i.e., σs2σ>902=0.202, where σ>90 denotes the uncertainty in Rrs explained by β˜p(γ90°). A constraint on χp was introduced in the fourth test, and, removing the uncertainties due to χp, the maximum uncertainty due to VSF shapes was reduced to 10% [Fig. 3(d)], i.e., σs2σχ2=0.102, where σχ denotes the uncertainty in Rrs explained by χp. Eq. (13) summarizes the results of these four tests:

{σs2=0.652σs2σBp2=0.352σs2σ>902=0.202σs2σχ2=0.102.

Solving Eq. (13), we have σBp=0.55, σ>90=0.62, and σχ=0.64. Therefore, among the total uncertainties in Rrs due to the shape of a VSF, about 71% (=σBp2/σs2) can be explained by Bp, 90% (=σ>902/σs2) by β˜p(γ90°), and 97% (=σχ2/σs2) by the χp factor.

From the perspective of ocean color remote sensing, it is difficult to measure Bp or β˜p(γ90°), which requires b, which unfortunately cannot be retrieved from ocean color, at least for now. On the other hand, both the QSS approximation [Eq. (11)] and the exact Rrs models, such as the one developed by Zaneveld [8,9], indicate that Rrs is directly proportional to the χ factor. Gordon [43] showed an example to retrieve the VSF shape, which was represented using an analytic equation first applied by Beardsley and Zaneveld [44]. Recently, Zhang et al. [45] found that the χp factor of a VSF can be represented by a linear mixing of two end members, the scattering by particles of sizes much smaller than the wavelength of light, and the scattering by particles of sizes much greater than the wavelength of light. The χp factors of both end members can be derived analytically. They also showed that the mixing ratio can be related to bbp, which can be retrieved from ocean color observation. However, this relationship is still preliminary and needs further validation. Looking forward, it remains to be tested whether the uncertainty in Rrs can be significantly constrained by directly accounting for the χ factor of a VSF.

Our analysis was conducted at 532 nm, at which the VSFs were measured. Based on Eq. (11), how the particle PFs affect the spectral variation of Rrs mainly depends on if and how the χp factor varies spectrally. While some studies found no spectral dependence in χp for oceanic particles [18] or phytoplankton cultures [46,47], Chang et al. [13] indicated that spectral differences could contribute to the up to 20% difference between simulated Lw and measured Lw, and Chami et al. [19] found χp varied spectrally by ±6% in a non-blooming coastal water but up to ±20% in algal cultures. How VSF shapes affect the spectral variation of Rrs remains to be studied.

Funding

National Science Foundation (NSF) (1458962); Directorate for Geosciences (GEO); National Aeronautics and Space Administration (NASA) (NNX13AN72G, NNX15AC85G); U.S. Naval Research Laboratory (NRL) (72-1C01); Office of Naval Research (ONR).

Acknowledgment

This study has benefited from discussion with Dr. Zhongping Lee. The comments by two anonymous reviewers and by Dr. Emmanuel Boss have greatly improved the paper.

REFERENCES

1. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988). [CrossRef]  

2. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II Bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993). [CrossRef]  

3. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996). [CrossRef]  

4. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975). [CrossRef]  

5. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977). [CrossRef]  

6. Z. Lee, K. L. Carder, and K. Du, “Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance,” Appl. Opt. 43, 4957–4964 (2004). [CrossRef]  

7. A. Morel and H. Loisel, “Apparent optical properties of oceanic water: dependence on the molecular scattering contribution,” Appl. Opt. 37, 4765–4776 (1998). [CrossRef]  

8. J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982). [CrossRef]  

9. J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. Oceans 100, 13135–13142 (1995). [CrossRef]  

10. W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54, 29–38 (2012). [CrossRef]  

11. Z. Lee, K. Du, K. J. Voss, G. Zibordi, B. Lubac, R. Arnone, and A. Weidemann, “An inherent-optical-property-centered approach to correct the angular effects in water-leaving radiance,” Appl. Opt. 50, 3155–3167 (2011). [CrossRef]  

12. B. Bulgarelli, G. Zibordi, and J.-F. Berthon, “Measured and modeled radiometric quantities in coastal waters: toward a closure,” Appl. Opt. 42, 5365–5381 (2003). [CrossRef]  

13. G. C. Chang, T. D. Dickey, C. D. Mobley, E. Boss, and W. S. Pegau, “Toward closure of upwelling radiance in coastal waters,” Appl. Opt. 42, 1574–1582 (2003). [CrossRef]  

14. I. Lefering, F. Bengil, C. Trees, R. Röttgers, D. Bowers, A. Nimmo-Smith, J. Schwarz, and D. McKee, “Optical closure in marine waters from in situ inherent optical property measurements,” Opt. Express 24, 14036–14052 (2016). [CrossRef]  

15. J. Pitarch, G. Volpe, S. Colella, R. Santoleri, and V. Brando, “Absorption correction and phase function shape effects on the closure of apparent optical properties,” Appl. Opt. 55, 8618–8636 (2016). [CrossRef]  

16. A. Tonizzo, M. Twardowski, S. McLean, K. Voss, M. Lewis, and C. Trees, “Closure and uncertainty assessment for ocean color reflectance using measured volume scattering functions and reflective tube absorption coefficients with novel correction for scattering,” Appl. Opt. 56, 130–146 (2017). [CrossRef]  

17. M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006). [CrossRef]  

18. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001). [CrossRef]  

19. M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006). [CrossRef]  

20. J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009). [CrossRef]  

21. X. Zhang, E. Boss, and D. J. Gray, “Significance of scattering by oceanic particles at angles around 120 degree,” Opt. Express 22, 31329–31336 (2014). [CrossRef]  

22. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express 17, 5698–5710 (2009). [CrossRef]  

23. A. Morel, “Light scattering in seawater: experimental results and theoretical approach,” in Optics of the Sea (Interface and In-water Transmission and Imaging) (AGARD, 1973).

24. H. R. Gordon, “Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment,” Appl. Opt. 32, 7505–7511 (1993). [CrossRef]  

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

26. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994). [CrossRef]  

27. G. R. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” Proc. SPIE 3761, 62–70 (1999). [CrossRef]  

28. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, 1972).

29. 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). [CrossRef]  

30. X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51, 5085–5099 (2012). [CrossRef]  

31. X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013). [CrossRef]  

32. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

33. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993). [CrossRef]  

34. V. I. Mankovsky and V. I. Haltrin, “Light scattering phase functions measured in waters of Mediterranean Sea,” in MTS/IEEE OCEANS (2002), Vol. 2364, pp. 2368–2373.

35. V. I. Mankovsky and V. I. Haltrin, “Phase functions of light scattering measured in waters of world ocean and lake Baykal,” in IEEE International Geoscience and Remote Sensing Symposium (2002), Vol. 3576, pp. 3570–3572.

36. X. Zhang and D. J. Gray, “Backscattering by very small particles in coastal waters,” J. Geophys. Res. Oceans 120, 6914–6926 (2015). [CrossRef]  

37. C. D. Mobley, “Hydrolight 5.2,” Technical Documentation (Sequoia Scientific, 2013).

38. IOCCG, “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications,” in Reports of the International Ocean-Colour Coordinating Group, Z.-P. Lee, ed. (IOCCG, 2006), Vol. 5.

39. G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985). [CrossRef]  

40. N. G. Jerlov, Marine Optics, Elsevier Oceanography Series (Elsevier, 1976), Vol. 14, p. 231.

41. H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973). [CrossRef]  

42. C. D. Mobley, E. Boss, and C. Roesler, “Ocean optics web book: the quasi-single-scattering approximation,” http://www.oceanopticsbook.info/view/radiative_transfer_theory/level_2/the_quasisinglescattering_approximation, retrieved June 20, 2017, 2011.

43. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989). [CrossRef]  

44. G. F. Beardsley and J. R. V. Zaneveld, “Theoretical dependence of the near-asymptotic apparent optical properties on the inherent optical properties of sea water*,” J. Opt. Soc. Am. 59, 373–377 (1969). [CrossRef]  

45. X. Zhang, G. R. Fournier, and D. J. Gray, “Interpretation of scattering by oceanic particles around 120 degrees and its implication in ocean color studies,” Opt. Express 25, A191–A199 (2017). [CrossRef]  

46. R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004). [CrossRef]  

47. A. L. Whitmire, W. S. Pegau, L. Karp-Boss, E. Boss, and T. J. Cowles, “Spectral backscattering properties of marine phytoplankton cultures,” Opt. Express 18, 15073–15093 (2010). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
    [Crossref]
  2. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II Bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
    [Crossref]
  3. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
    [Crossref]
  4. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
    [Crossref]
  5. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [Crossref]
  6. Z. Lee, K. L. Carder, and K. Du, “Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance,” Appl. Opt. 43, 4957–4964 (2004).
    [Crossref]
  7. A. Morel and H. Loisel, “Apparent optical properties of oceanic water: dependence on the molecular scattering contribution,” Appl. Opt. 37, 4765–4776 (1998).
    [Crossref]
  8. J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982).
    [Crossref]
  9. J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. Oceans 100, 13135–13142 (1995).
    [Crossref]
  10. W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54, 29–38 (2012).
    [Crossref]
  11. Z. Lee, K. Du, K. J. Voss, G. Zibordi, B. Lubac, R. Arnone, and A. Weidemann, “An inherent-optical-property-centered approach to correct the angular effects in water-leaving radiance,” Appl. Opt. 50, 3155–3167 (2011).
    [Crossref]
  12. B. Bulgarelli, G. Zibordi, and J.-F. Berthon, “Measured and modeled radiometric quantities in coastal waters: toward a closure,” Appl. Opt. 42, 5365–5381 (2003).
    [Crossref]
  13. G. C. Chang, T. D. Dickey, C. D. Mobley, E. Boss, and W. S. Pegau, “Toward closure of upwelling radiance in coastal waters,” Appl. Opt. 42, 1574–1582 (2003).
    [Crossref]
  14. I. Lefering, F. Bengil, C. Trees, R. Röttgers, D. Bowers, A. Nimmo-Smith, J. Schwarz, and D. McKee, “Optical closure in marine waters from in situ inherent optical property measurements,” Opt. Express 24, 14036–14052 (2016).
    [Crossref]
  15. J. Pitarch, G. Volpe, S. Colella, R. Santoleri, and V. Brando, “Absorption correction and phase function shape effects on the closure of apparent optical properties,” Appl. Opt. 55, 8618–8636 (2016).
    [Crossref]
  16. A. Tonizzo, M. Twardowski, S. McLean, K. Voss, M. Lewis, and C. Trees, “Closure and uncertainty assessment for ocean color reflectance using measured volume scattering functions and reflective tube absorption coefficients with novel correction for scattering,” Appl. Opt. 56, 130–146 (2017).
    [Crossref]
  17. M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
    [Crossref]
  18. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
    [Crossref]
  19. M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
    [Crossref]
  20. J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009).
    [Crossref]
  21. X. Zhang, E. Boss, and D. J. Gray, “Significance of scattering by oceanic particles at angles around 120 degree,” Opt. Express 22, 31329–31336 (2014).
    [Crossref]
  22. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express 17, 5698–5710 (2009).
    [Crossref]
  23. A. Morel, “Light scattering in seawater: experimental results and theoretical approach,” in Optics of the Sea (Interface and In-water Transmission and Imaging) (AGARD, 1973).
  24. H. R. Gordon, “Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment,” Appl. Opt. 32, 7505–7511 (1993).
    [Crossref]
  25. C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002).
    [Crossref]
  26. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
    [Crossref]
  27. G. R. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” Proc. SPIE 3761, 62–70 (1999).
    [Crossref]
  28. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, 1972).
  29. 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).
    [Crossref]
  30. X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51, 5085–5099 (2012).
    [Crossref]
  31. X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
    [Crossref]
  32. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  33. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [Crossref]
  34. V. I. Mankovsky and V. I. Haltrin, “Light scattering phase functions measured in waters of Mediterranean Sea,” in MTS/IEEE OCEANS (2002), Vol. 2364, pp. 2368–2373.
  35. V. I. Mankovsky and V. I. Haltrin, “Phase functions of light scattering measured in waters of world ocean and lake Baykal,” in IEEE International Geoscience and Remote Sensing Symposium (2002), Vol. 3576, pp. 3570–3572.
  36. X. Zhang and D. J. Gray, “Backscattering by very small particles in coastal waters,” J. Geophys. Res. Oceans 120, 6914–6926 (2015).
    [Crossref]
  37. C. D. Mobley, “Hydrolight 5.2,” Technical Documentation (Sequoia Scientific, 2013).
  38. IOCCG, “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications,” in Reports of the International Ocean-Colour Coordinating Group, Z.-P. Lee, ed. (IOCCG, 2006), Vol. 5.
  39. G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985).
    [Crossref]
  40. N. G. Jerlov, Marine Optics, Elsevier Oceanography Series (Elsevier, 1976), Vol. 14, p. 231.
  41. H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973).
    [Crossref]
  42. C. D. Mobley, E. Boss, and C. Roesler, “Ocean optics web book: the quasi-single-scattering approximation,” http://www.oceanopticsbook.info/view/radiative_transfer_theory/level_2/the_quasisinglescattering_approximation , retrieved June 20, 2017, 2011.
  43. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [Crossref]
  44. G. F. Beardsley and J. R. V. Zaneveld, “Theoretical dependence of the near-asymptotic apparent optical properties on the inherent optical properties of sea water*,” J. Opt. Soc. Am. 59, 373–377 (1969).
    [Crossref]
  45. X. Zhang, G. R. Fournier, and D. J. Gray, “Interpretation of scattering by oceanic particles around 120 degrees and its implication in ocean color studies,” Opt. Express 25, A191–A199 (2017).
    [Crossref]
  46. R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
    [Crossref]
  47. A. L. Whitmire, W. S. Pegau, L. Karp-Boss, E. Boss, and T. J. Cowles, “Spectral backscattering properties of marine phytoplankton cultures,” Opt. Express 18, 15073–15093 (2010).
    [Crossref]

2017 (2)

2016 (2)

2015 (1)

X. Zhang and D. J. Gray, “Backscattering by very small particles in coastal waters,” J. Geophys. Res. Oceans 120, 6914–6926 (2015).
[Crossref]

2014 (1)

2013 (1)

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
[Crossref]

2012 (2)

2011 (1)

2010 (1)

2009 (2)

2006 (3)

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

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).
[Crossref]

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

2004 (2)

Z. Lee, K. L. Carder, and K. Du, “Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance,” Appl. Opt. 43, 4957–4964 (2004).
[Crossref]

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
[Crossref]

2003 (2)

2002 (1)

2001 (1)

1999 (1)

G. R. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” Proc. SPIE 3761, 62–70 (1999).
[Crossref]

1998 (1)

1996 (1)

1995 (1)

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. Oceans 100, 13135–13142 (1995).
[Crossref]

1994 (1)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

1993 (3)

1989 (1)

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

1988 (1)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

1985 (1)

G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985).
[Crossref]

1982 (1)

1977 (1)

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

1975 (1)

1973 (1)

1969 (1)

Ahmad, Z.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

Arnone, R.

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

Balch, W. M.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
[Crossref]

Beardsley, G. F.

Bengil, F.

Berthon, J.-F.

Bi, L.

Boss, E.

Bowers, D.

Brando, V.

Brown, C. W.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
[Crossref]

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[Crossref]

Bulgarelli, B.

Carder, K. L.

Chami, M.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

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).
[Crossref]

Chang, G. C.

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

Colella, S.

Cowles, T. J.

Dickey, T. D.

Du, K.

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

Fournier, G. R.

X. Zhang, G. R. Fournier, and D. J. Gray, “Interpretation of scattering by oceanic particles around 120 degrees and its implication in ocean color studies,” Opt. Express 25, A191–A199 (2017).
[Crossref]

G. R. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” Proc. SPIE 3761, 62–70 (1999).
[Crossref]

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

Freda, W.

W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54, 29–38 (2012).
[Crossref]

Gallegos, C. L.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

Gentili, B.

Gordon, H. R.

Gray, D. J.

Guillard, R. R. L.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
[Crossref]

Haltrin, V. I.

V. I. Mankovsky and V. I. Haltrin, “Light scattering phase functions measured in waters of Mediterranean Sea,” in MTS/IEEE OCEANS (2002), Vol. 2364, pp. 2368–2373.

V. I. Mankovsky and V. I. Haltrin, “Phase functions of light scattering measured in waters of world ocean and lake Baykal,” in IEEE International Geoscience and Remote Sensing Symposium (2002), Vol. 3576, pp. 3570–3572.

Harding, L. W.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

He, M.-X.

Herman, J. R.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

Hu, L.

Humphreys, T. J.

G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985).
[Crossref]

Huot, Y.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51, 5085–5099 (2012).
[Crossref]

Jacobs, M. M.

Jerlov, N. G.

N. G. Jerlov, Marine Optics, Elsevier Oceanography Series (Elsevier, 1976), Vol. 14, p. 231.

Jin, Z.

Jonasz, M.

G. R. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” Proc. SPIE 3761, 62–70 (1999).
[Crossref]

Karp-Boss, L.

Kattawar, G. W.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[Crossref]

G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985).
[Crossref]

Khomenko, G.

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).
[Crossref]

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

Korotaev, G.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

Lee, Z.

Lefering, I.

Lewis, M.

Leymarie, E.

Loisel, H.

Lubac, B.

Mankovsky, V. I.

V. I. Mankovsky and V. I. Haltrin, “Phase functions of light scattering measured in waters of world ocean and lake Baykal,” in IEEE International Geoscience and Remote Sensing Symposium (2002), Vol. 3576, pp. 3570–3572.

V. I. Mankovsky and V. I. Haltrin, “Light scattering phase functions measured in waters of Mediterranean Sea,” in MTS/IEEE OCEANS (2002), Vol. 2364, pp. 2368–2373.

Marken, E.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

McKee, D.

McLean, S.

Mobley, C. D.

Morel, A.

Neale, P. J.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

Nimmo-Smith, A.

Pegau, W. S.

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, 1972).

Piskozub, J.

W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54, 29–38 (2012).
[Crossref]

Pitarch, J.

Plass, G. N.

G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985).
[Crossref]

Prieur, L.

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

Reinersman, P.

Rhea, W. J.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
[Crossref]

Röttgers, R.

Santoleri, R.

Schwarz, J.

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

Stamnes, J. J.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

Stamnes, K.

Stavn, R. H.

Subramaniam, A.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

Sullivan, J. M.

Sundman, L. K.

Tonizzo, A.

Trees, C.

Twardowski, M.

Twardowski, M. S.

Tzortziou, M.

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

Vaillancourt, R. D.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
[Crossref]

Volpe, G.

Voss, K.

Voss, K. J.

Weidemann, A.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
[Crossref]

Z. Lee, K. Du, K. J. Voss, G. Zibordi, B. Lubac, R. Arnone, and A. Weidemann, “An inherent-optical-property-centered approach to correct the angular effects in water-leaving radiance,” Appl. Opt. 50, 3155–3167 (2011).
[Crossref]

Whitmire, A. L.

You, Y.

Zaneveld, J. R. V.

Zhang, X.

Zibordi, G.

Appl. Opt. (19)

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. II Bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
[Crossref]

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
[Crossref]

H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[Crossref]

Z. Lee, K. L. Carder, and K. Du, “Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance,” Appl. Opt. 43, 4957–4964 (2004).
[Crossref]

A. Morel and H. Loisel, “Apparent optical properties of oceanic water: dependence on the molecular scattering contribution,” Appl. Opt. 37, 4765–4776 (1998).
[Crossref]

J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982).
[Crossref]

Z. Lee, K. Du, K. J. Voss, G. Zibordi, B. Lubac, R. Arnone, and A. Weidemann, “An inherent-optical-property-centered approach to correct the angular effects in water-leaving radiance,” Appl. Opt. 50, 3155–3167 (2011).
[Crossref]

B. Bulgarelli, G. Zibordi, and J.-F. Berthon, “Measured and modeled radiometric quantities in coastal waters: toward a closure,” Appl. Opt. 42, 5365–5381 (2003).
[Crossref]

G. C. Chang, T. D. Dickey, C. D. Mobley, E. Boss, and W. S. Pegau, “Toward closure of upwelling radiance in coastal waters,” Appl. Opt. 42, 1574–1582 (2003).
[Crossref]

J. Pitarch, G. Volpe, S. Colella, R. Santoleri, and V. Brando, “Absorption correction and phase function shape effects on the closure of apparent optical properties,” Appl. Opt. 55, 8618–8636 (2016).
[Crossref]

A. Tonizzo, M. Twardowski, S. McLean, K. Voss, M. Lewis, and C. Trees, “Closure and uncertainty assessment for ocean color reflectance using measured volume scattering functions and reflective tube absorption coefficients with novel correction for scattering,” Appl. Opt. 56, 130–146 (2017).
[Crossref]

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
[Crossref]

J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009).
[Crossref]

H. R. Gordon, “Sensitivity of radiative transfer to small-angle scattering in the ocean: quantitative assessment,” Appl. Opt. 32, 7505–7511 (1993).
[Crossref]

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

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).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51, 5085–5099 (2012).
[Crossref]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[Crossref]

H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973).
[Crossref]

Biogeosciences (1)

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
[Crossref]

Estuarine Coastal Shelf Sci. (1)

M. Tzortziou, J. R. Herman, C. L. Gallegos, P. J. Neale, A. Subramaniam, L. W. Harding, and Z. Ahmad, “Bio-optics of the Chesapeake Bay from measurements and radiative transfer closure,” Estuarine Coastal Shelf Sci. 68, 348–362 (2006).
[Crossref]

J. Geophys. Res. Atmos. (1)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. Atmos. 93, 10909–10924 (1988).
[Crossref]

J. Geophys. Res. Oceans (4)

G. N. Plass, G. W. Kattawar, and T. J. Humphreys, “Influence of the oceanic scattering phase function on the radiance,” J. Geophys. Res. Oceans 90, 3347–3351 (1985).
[Crossref]

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. Oceans 111, C05013 (2006).
[Crossref]

J. R. V. Zaneveld, “A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties,” J. Geophys. Res. Oceans 100, 13135–13142 (1995).
[Crossref]

X. Zhang and D. J. Gray, “Backscattering by very small particles in coastal waters,” J. Geophys. Res. Oceans 120, 6914–6926 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

J. Plankton Res. (1)

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26, 191–212 (2004).
[Crossref]

Limnol. Oceanogr. (2)

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

Oceanologia (1)

W. Freda and J. Piskozub, “Revisiting the role of oceanic phase function in remote sensing reflectance,” Oceanologia 54, 29–38 (2012).
[Crossref]

Opt. Express (5)

Proc. SPIE (2)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

G. R. Fournier and M. Jonasz, “Computer-based underwater imaging analysis,” Proc. SPIE 3761, 62–70 (1999).
[Crossref]

Other (9)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, 1972).

A. Morel, “Light scattering in seawater: experimental results and theoretical approach,” in Optics of the Sea (Interface and In-water Transmission and Imaging) (AGARD, 1973).

C. D. Mobley, “Hydrolight 5.2,” Technical Documentation (Sequoia Scientific, 2013).

IOCCG, “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications,” in Reports of the International Ocean-Colour Coordinating Group, Z.-P. Lee, ed. (IOCCG, 2006), Vol. 5.

V. I. Mankovsky and V. I. Haltrin, “Light scattering phase functions measured in waters of Mediterranean Sea,” in MTS/IEEE OCEANS (2002), Vol. 2364, pp. 2368–2373.

V. I. Mankovsky and V. I. Haltrin, “Phase functions of light scattering measured in waters of world ocean and lake Baykal,” in IEEE International Geoscience and Remote Sensing Symposium (2002), Vol. 3576, pp. 3570–3572.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

N. G. Jerlov, Marine Optics, Elsevier Oceanography Series (Elsevier, 1976), Vol. 14, p. 231.

C. D. Mobley, E. Boss, and C. Roesler, “Ocean optics web book: the quasi-single-scattering approximation,” http://www.oceanopticsbook.info/view/radiative_transfer_theory/level_2/the_quasisinglescattering_approximation , retrieved June 20, 2017, 2011.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. (a) Comparison of particle PFs derived from: our measurements (gray lines); samples of historical Petzold’s measurements in clear water of the Tongue of the Ocean (black dotted line), in California offshore coastal water (black dashed line), and in very turbid water of San Diego harbor (black dotted-dashed line) [32]; the average particle PF of Petzold’s measurements in San Diego harbor [33] (black solid line); the analytical Fournier–Forand (FF) formula having a B p = 1.83 % [25] (blue dashed line); and a series of particle PFs generated by linear mixing of B p = 1 % FF PF and the average Petzold PF, as used in Lee et al. [11] (green solid lines). The red line is one of the measured particle PFs referred to in Fig. 5. To highlight the variations, the x axes are in logarithmic scale for angles < 30 ° and in linear scales for larger angles. (b) Corresponding χ p factors. (c) Histogram of B p calculated from measured particle PFs.
Fig. 2.
Fig. 2. R rs ( θ s = 30 ° , θ v = 0 ° ) simulated with measured particle phase functions (color coded by the measurement locations) are shown as a function of b b / ( a + b b ) .
Fig. 3.
Fig. 3. Percentage difference between R rs ( θ s = 30 ° , θ v = 0 ° ) simulated with two different particle phase functions. The whiskers boxes, showing the minimum, lower quartile, median, upper quartile, and maximum values, summarize the differences at the selected b b / ( a + b b ) values. (a) Simulations performed with same b b p values. (b) Simulations performed with same b b p values and phase functions having the same B p values. The dotted line at 10% represents the maximum difference found in Ref. [25], the dashed line at 20% represents the maximum difference found in Ref. [29], and the dotted-dashed line at 40% represents the maximum difference found in Ref. [16]. (c) Simulations performed with same b b p values and very similar β ˜ p ( γ 90 ° ) (logarithmic value within 1%). (d) Simulations performed with same b b p values and very similar χ p factor (within 1%).
Fig. 4.
Fig. 4. Distribution of the maximum value of d R rs . In the polar plots, ao-56-24-6881-i001 indicates θ s , radial direction represents θ v , and polar direction represents ϕ . Column (a) results with same b b p ; column (b) results with same b b p and B p ; column (c) results with same b b p and similar β ˜ p ( γ 90 ° ) (logarithmic value within 1%); column (d) results with same b b p and similar χ p factor (within 1%).
Fig. 5.
Fig. 5. Comparison between R rs simulated with one of the measured phase functions (red line in Fig. 1) by HydroLight and R rs estimated using the Lee’s et al. [11] model at all viewing geometries ( θ s = 0 75 ° , θ v = 0 70 ° , and ϕ = 0 180 ° ).

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

R rs ( λ , θ s , θ v , φ ) = L w ( λ , θ s , θ v , φ ) E d ( λ , θ s ) ,
R rs = f Q b b a + b b .
b b = 2 π π / 2 π β ( γ ) sin γ d γ ,
β ˜ ( γ ) = β ( γ ) b ,
B = b b b = 2 π π / 2 π β ˜ ( γ ) sin γ d γ ,
χ ( γ 90 ° ) = b b 2 π β ( γ ) = B 2 π β ˜ ( γ ) .
β p ( γ ) = A γ m ,
b p = 2 π 2 m β p ( γ 1 ) γ 1 2 + 2 π 0.1 π β p ( γ ) sin γ d γ .
d R rs ( a , b b , θ s , θ v , φ ) = | R rs ( i , a , b b , θ s , θ v , φ ) R rs ( j , a , b b , θ s , θ v , φ ) | 1 2 [ R rs ( i , a , b b , θ s , θ v , φ ) + R rs ( j , a , b b , θ s , θ v , φ ) ] × 100 % ,
R rs ( θ , ϕ , θ s ) = 1 cos θ v + cos θ s β ( γ s ) a + b b ,
R rs ( θ , ϕ , θ s ) = 1 2 π χ ( γ s ) 1 cos θ v + cos θ s b b a + b b .
R rs ( θ , ϕ , θ s ) = β ˜ ( γ s ) B 1 cos θ v + cos θ s b b a + b b .
{ σ s 2 = 0.65 2 σ s 2 σ B p 2 = 0.35 2 σ s 2 σ > 90 2 = 0.20 2 σ s 2 σ χ 2 = 0.10 2 .

Metrics