Vector radiative transfer model for coupled atmosphere and ocean systems including inelastic sources in ocean waters

PENG-WANG ZHAI,1,* YONGXIANG HU,2 DAVID M. WINKER,2 BRYAN A. FRANZ, 3 JEREMY WERDELL, 3 AND EMMANUEL BOSS4 1Department of Physics, University of Maryland Baltimore County, Baltimore, MD, 21250, USA 2MS 475 NASA Langley Research Center, Hampton, VA 23681-2199, USA 3NASA Goddard Space Flight Center, Code 616, Greenbelt, Maryland 20771, USA 4School of Marine Sciences, University of Maine, Orono, Maine 04401, USA *pwzhai@umbc.edu


Introduction
Radiative transfer for coupled atmosphere and ocean systems (CAOS) combines atmospheric radiation and ocean optics in a unified theoretical frame [1,2].Exact radiative transfer solutions preserve the polarization nature of the radiation, which is generally referred to as vector radiative transfer [3][4][5][6][7][8][9][10][11][12][13][14][15].If the polarization nature of the radiation is ignored, the vector radiative transfer is reduced to the scalar radiative transfer, which only solves the approximated intensity of the radiation [16][17][18][19].Sea surface roughness is normally parameterized in terms of wind speed at 10 meters above ocean surface [20].Recently new developments have been made to study the reflection and transmission properties of rough surfaces with different Fourier spectra, sea swell, and diffuse incident light fields [21][22][23].
In this paper we report an exact vector radiative transfer model for CAOS that incorporates both elastic and inelastic scattering processes.The solution is based on the successive order of scattering (SOS) method [11,12].Previously the implementation of Raman scattering in the SOS for CAOS has been published [47].The new addition to the SOS model is the fluorescence by colored dissolved organic matter (FDOM) and chlorophyll fluorescence, which is the main focus of this paper.Our model possesses a unique combination of the following features: 1) full atmosphere and ocean coupling; 2) exact treatment of the polarization; 3) inclusion of all major inelastic scattering mechanisms; 4) formulas based on the SOS method that are efficient and accurate; 5) no approximation on the angular and polarization distribution features of the elastic and inelastic scattering sources.
The FDOM and chlorophyll fluorescence are assumed to be isotropic and non-polarized [1].However, it is still necessary to incorporate fluorescence in a vector radiative transfer model because the multiple interaction between fluorescence, elastic, and Raman scattering induces polarization.This paper is organized as follows: Sec. 2 presents the generalized vector inelastic radiative transfer equation based on the scalar version in [1]; Sec. 3 outlines the ocean biooptical model used for simulation; Sec. 4 shows the practical simulation procedures and results; Sec. 5 summarizes the main conclusions.

Inelastic vector radiative transfer equation
The inelastic vector radiative transfer equation for a plane-parallel system without thermal emission can be written as [1,32,47]: where L=[I, Q, U, V] T is the Stokes vector and superscript T stands for transpose; S and S i are the elastic and inelastic source terms, respectively; z is the vertical coordinate; µ 0 = cos θ 0 ; µ = cos θ; θ 0 and θ are the solar and viewing zenith angles, respectively; φ is the viewing azimuth angle; λ is the emission wavelength; c is the extinction coefficient.The inelastic source terms are summation of three mechanisms: where the subscripts R, Y, and C stand for Raman scattering, FDOM, and chlorophyll fluorescense, respectively.
The elastic source S and Raman scattering source S R have been described in [47] with the same terminology convention, which we omit here for conciseness.The readers are referred to [47] for a detailed description.Here we focus on S Y and S C in the theoretical frame of vector radiative transfer: where the subscript F can be either Y or C, denoting FDOM or chlorophyll; b F is the fluorescence scattering coefficient; and λ e denotes the excitation wavelength.We assume the fluorescence phase matrix P F is isotropic and unpolarized [1]: where P F,i j is the i jth matrix element of P F .The fluorescence scattering coefficient b F can be written as a product of two terms [1]: where a F is the absorption coefficient of either colored dissolved organic matter (CDOM) or chlorophyll at excitation wavelength λ e ; and f F is the fluorescence redistribution function [1]: where η F (z, λ, λ e ) is the spectral fluorescence quantum efficiency function, defined as the ratio of the number of photons emitted at λ per unit wavelength interval and the number of photons absorbed at λ e .The factor λ e /λ accounts for energy being proportional to 1/λ for a photon.The function η F varies with substance.For FDOM, we replace subscript F with Y, and use the parameterization by Hawes et al. [49]: where A 0 , A 1 , A 2 , B 1 , and B 2 are parameters that depend on the particular composition of the yellow matter.Hawes et al [49] provided tabulated values of these parameters for four fulvic acid samples: (FA7, FA8, FA9, FA10); and seven humic acid samples: (HA1, HA2, HA3, HA4, HA5, HA6), extracted from ocean waters.In this work we model FDOM using a mixture of 9/10 of FA7 and 1/10 of HA6, which was also assumed in [49].The symbol g Y is a dimensionless function, which specifies the range of excitation wavelength within which η Y is significant: However, the inherent optical property measurement for wavelengths below 350 nm is less common and difficult to verify.Hereafter the wavelengths λ and λ e are referred in the unit of nm.
We have assumed g Y = 0 for λ e <350 nm.This can be justified with the strong ozone absorption and decreased solar spectral irradiance in the UV. Green and Blough [38] found that the FDOM fluorescence quantum efficiency is different when FDOM is extracted or still dissolved in water samples.This mitigates the applicability of the data from Hawes et al. [49], which was measured from the FDOM samples extracted from ocean waters.Readers are referred to [38] for a detailed discussion of the measurement of FDOM in natural waters.However, the data in [49] are still the only complete set of data that covers a whole range of excitation and emission wavelengths.We better adopt the approach of [49] in this work to demonstrate our implementation of FDOM fluorescence.When future data are available, they can be easily adopted within the computational package.
The chlorophyll fluorescence quantum efficiency function η C takes the form of [1]: where Φ C (λ) is the fluorescence quantum yield of chlorophyll: the dimensionless function g C specifies the spectral range: The emission function h C (λ) is approximated by a Gaussian [1]: where λ C,0 =685 nm is the wavelength of maximum emission, and σ C = 10.6 nm is the standard deviation of the Gaussian.The quantum yield Φ C typically ranges between 0.01 and 0.05 , which depends on phytoplankton species, physiological state, and other environmental factors [1].One mechanism of Φ C variation is called non-photochemical quenching (NPQ), under which algae reduce Φ C under large irradiance to protect themselves [43,50].The impacts of NPQ to chlorophyll fluorescence is beyond the scope of this study.We used a constant Φ C =0.02 to show the mean features of chlorophyll fluorescence.

Biooptical models of ocean waters
Inelastic scattering simulations need to consider transspectral optical properties, for which a biooptical model for ocean waters is necessary.The set of biooptical models used in [47,51] are followed with modifications highlighted below in order to better conform spectral behavior of particle absorption and scattering coefficients.The ocean water is composed of three components: pure ocean water, phytoplankton covariant particles, and absorptive CDOM.The total extinction coefficient c(λ, [Chla]) at wavelength λ for a chlorophyll a value [Chla] is assumed to follow: where the subscript w, p, and Y stand for pure sea water, phytoplankton covariant particles, and CDOM, respectively.The extinction coefficient for each component is a sum of the absorption and scattering coefficient [1]: The pure water absorption coefficient a w are based on data in [52]; and b w follows: [1,53]: The dependence of b w on salinity and temperature is not considered here.The extinction coefficient c p of phytoplankton covariant particles takes the form of [54,55]: where both p 1 and R 1 are numbers that can vary in nature and are in general unknown.In [55], p 1 is modeled as a random number uniformly distributed between 0.06 and 0.6 and R 1 is modeled as a random number uniformly distributed between 0 and 1.In this work we assigned p 1 = 0.33 and R 1 = 0.5 for the sake of clarity of the simulations.The particle absorption coefficients a p ([Chla],λ) are a function [Chla] and wavelength λ [56].As noted in [47], the absorption data in [56] covers the wavelength range of 400 nm to 700 nm.In order to evaluate the inelastic source term, a second set of the absorption data is used to expand the spectral range to the UV [57].The continuity of these two sets of absorption data is assured by scaling the data in [57] to the same value as in [56] at 440 nm.The particle scattering coefficient is simply the difference between c p and a p : b p = c p − a p .We assume the scattering coefficient for CDOM b Y = 0.The CDOM absorption is [55,[58][59][60]: where R 2 is a random number uniformly distributed between 0 and 1 to account for natural variability as introduced by its original authors.In this paper we assigned R 2 = 0.5.It should be noted that the origin of CDOM is from diverse sources including natural and industrial land discharges, and the biological activity averaged over a long period [58].Consequently, the correlation between CDOM and [Chla] is rather weak for measurements taken from specific time and locations.However, the CDOM and chlorophyll a concentrations averaged over large spatial and time scales do correlate [59,60].Here we adopt Eq. ( 17) to reflect this correlation in our model with the understanding that the correlation may not be applicable to local measurements [55].In this situation, we could use local measurements of a Y (440) to replace a Y (440, [Chla]) in the formulas, with other parts intact.
As an illustration, Fig. 1 shows the single scattering albedo for ocean waters determined from the absorption and scattering properties outlined above for a set of [Chla] values.The spectral shape agrees with the literature [9], which demonstrates the consistency of the bio-optical model employed.( For a specific value of [Chla], Eq. ( 18) is used to determine B b p .The elastic scattering phase function P 11, p for phytoplankton covariant particles is then the Fournier-Forand phase function [62,63], parameterized based on backscattering ratio [64].The total ocean scattering phase function is a weighted sum of phytoplankton particle and pure water: where P 11,w (Θ) is the pure water phase function [1]; and Θ is the scattering angle.The elastic reduced scattering matrix of the ocean water is a parameterization of measurement by Voss [65,66].The main rationale of this choice is that the average measurements [65] represent the reality of a variety of ocean waters.In addition, we wish to preserve the features P 22 /P 11 observed in ocean waters, which is quite important in some cases.Note that the use of Mie theory will result in P 22 /P 11 = 1, which is not observed for ocean waters.

Simulations and discussion
Inelastic radiative transfer simulation involves two steps.The first step is to perform a series of elastic radiative transfer calculations at a range of excitation wavelengths that have notable contributions to the inelastic source term S i .The second step is to gather all the inelastic source terms S i and the elastic source S and perform the radiative transfer simulation at the emission wavelength.Higher order inelastic scattering terms of fluorescence being absorbed and re-emitted are proportional to the higher polynomial order of quantum yields which are assumed negligible and ignored in this work.The interaction of excitation and emission radiation fields requires an optical model that correlates the single scattering properties at different wavelengths for both the atmosphere and ocean.For the atmosphere, it is assumed to be a mixture of molecules and aerosols.The vertical profile of the molecule number density is determined by the 1976 US standard atmosphere [67].The molecule scattering optical depth is calculated using the method in [68].The molecule scattering matrix is the Rayleigh scattering matrix with a depolarization of 0.0284 [69].The aerosol model is the Maritime aerosol model with relative humidity of 80% [70].The aerosol optical depth at 550 nm is 0.15.The aerosol vertical distribution follows the average height distribution in [71].Gas absorption in the atmosphere is not considered, but rather assumed to be removed through atmospheric correction.The Cox-Munk model is used to specify ocean surface roughness based on wind speed [20].The dependence of ocean roughness on wind direction is neglected.The solar spectral irradiance data are those reported in [72].
The ocean optical properties are determined from the biooptical model in Sec. 3 for specified wavelengths and [Chla] values.The vertical distribution of [Chla] is a generalized Gaussian superposed onto a linearly decreasing background [73].To enable assigning a full vertical [Chla] profile to a single surface value, we first determine an integrated value over the euphotic zone through [73]: where the fitting parameters A and B are provided in Table 4 in [73].Then the depth of the euphotic layer Z eu is [74]: The depth z and chlorophyll a concentration [Chla] is then rescaled [73]: Given the above notations, the rescaled chlorophyll a concentration profile is [73]: where χ b is the rescaled background [Chla] value, s is the linear slope, χ ma x is the rescaled maximum concentration, ζ ma x is the depth at which maximum concentration is located; and, ∆ζ determines the width of the Gaussian peak.In this study, we have used the following parameters: χ b = 0.471, s=0.135, χ ma x =1.572, ζ ma x =0.969, and ∆ζ=0.393,which is the tropic class S1 case shown in Table 5 in [73].The final vertical [Chla] profile is given by: so that the vertical chlorophyll profile is conveniently determined by a specified surface value [Chla] surf .Because of the non-zero slope s, at some depth the formulas above will result in negative [Chla] values.When this happens, we just reset the [Chla] value at that depth to be zero.
Figure 2 shows the vertical [Chla] profiles for a set of [Chla] surf values: 0.03, 0.1, 0.3, 1.0, and 3.0 mg/m 3 , which are used in the following simulations.
The procedure to determine the Raman excitation wavelength range is described in [47].For FDOM and chlorophyll fluorescence simulations, the excitation wavelength range is specified by Eqs.(8) or (11).A spectral resolution of 10 nm is used to ensure accuracy while maintaining efficiency [46].Once all the radiation fields at the excitation wavelengths are finished, a final elastic vector radiative transfer simulation is performed with the inelastic source included.In [Chla] surf =0.03mg m -3 [Chla] surf =0.1mg m -3 [Chla] surf =0.3mg m -3 [Chla] surf =1mg m -3 [Chla] surf =3mg m -3 Fig. 2. Chlorophyll concentration profile for a set of [Chla] surf values the following results, the water column depth is fixed at 100 m with detectors evenly placed in the water column.This large depth leads to very large optical depths at some wavelengths due to large ocean water absorption coefficient.This is the reason that it takes around 30 minutes on average to simulate the angular and depth resolved inelastic scattering contribution at an emission wavelength with the gfortran compiler 5.0 an AMD Opteron(TM) 6386 SE Processor.
The solar zenith angle is 45 degrees and the wind speed is 5 m/s.The emission wavelengths are 412, 443, 490, 555, 665, 670, 675, 680, 685, 690 nm, in which those shorter than 670 nm are typical wavelengths used for ocean color remote sensing, while those longer than 670 nm are added to study the sensitivity of chlorophyll a fluorescence.We realize that there are some water vapor absorptions in those longer wavelengths, which is likely to change the relative magnitude of elastic and inelastic scattering signals.For the wavelengths of 680, 685, and 690 nm, we have calculated the water vapor transmission for a typical tropical atmospheric column, which represents an atmospheric condition with large water vapor absorption.A spectral filter of 2 nm resolution is used to smooth out the transmission spectra.The Atmospheric Radiative Transfer Simulator (ARTS) [75] is used for the calculation with the molecule absorption line parameters provided by the high-resolution transmission molecular absorption database (HITRAN) [76].It is estimated that the column transmission due to water vapor absorption at 680, 685, and 690 nm are smaller than 1%, which should not pose any noticeable impacts to the simulations presented in this paper.Recall that the focus of this study is the inelastic scattering in ocean waters and their contribution to the underwater polarized radiance field.

Radiance field
Figure 3 shows the inelastic scattering contribution to the total radiance field as a function of wavelength for a detector just below the ocean surface.The left column shows the absolute reflectance contribution L F,u,0 − /E t,0 − , where L is the radiance, the same quantity as I in the Stokes parameter; E t,0 − is the downwelling irradiance with 0 − shows the location of the detector, which is just below the ocean surface; and, subscript t stands for "total" radiation field; the subscript F can be either R, Y, or C, standing for Raman, FDOM, and or chlorophyll fluorescence; the subscript u denotes the upwelling radiance with the viewing zenith angle of zero.The right column shows the percentage fraction of inelastic scattering in the total upwelling nadir-viewing radiance 100 • L F,u,0 − /L t,u,0 − .Figure 3(a) shows that the Raman scattering reflectance peak location shifts from 490 nm to 555 nm as the [Chla] surf value increases.Figure 3(b) is similar to Fig. 3b in [47], except that the total radiance now includes FDOM and chlorophyll fluorescence.Raman scattering reflectance decreases as wavelength increases after the spectral peak.However, the percentage fraction remains largely flat because the total radiance field also decreases at similar rates.Consistent with the cases reported in [47], Raman scattering contributions decrease as [Chla] surf increases.The spectral trough centered at 685 nm in Fig. 3(b) is due to the large chlorophyll fluorescence contribution in that wavelength range, which can be observed in Figs.3(e) and 3(f).
The spectral shape of the fractional contribution in Fig. 3(d) is similar to the semianalytical results shown in [49].The order of magnitude is also consistent with the model result in [48] (See Fig. 3 of [48]).The absolute contribution in Fig. 3(c) largely resembles the spectral shape of Eq. ( 7) for sample HA7 reported in [49].These observations confirm that our implementation is consistent with the physics and generates reasonable results.The FDOM absolute contribution increases as [Chla] surf increases, with the largest contribution around 500 nm.The relative fraction ranges from 0 to 15% depending on the wavelength and [Chla] surf values.Figure 3 indicate the contribution of Raman scattering is larger than that of FDOM in the visible for [Chla] surf <1 mg m −3 .
For the selected wavelengths, the contribution of chlorophyll fluorescence is significant only at wavelengths between 665 nm and 690 nm.Figures 3(e) and 3(f) show that the chlorophyll fluorescence spectral shape is a Gaussian (see Eq. ( 12)) centered at 685 nm.The percentage fraction increases as [Chla] surf increases.The peak value changes from 30% for [Chla] surf =0.03 mg/m 3 to more than 60% for [Chla] surf =3 mg/m 3 .This indicates that the chlorophyll fluorescence band centered at 685 nm can be a source of information for ocean productivity remote sensing.Figure 4 provides some sense of the angular distribution of the inelastic fraction 100 • L R+Y+C, 0 − /L t, 0 − for two values of [Chla] surf : 0.03 mg m −3 and 3 mg m −3 .The subscript R+Y+C shows all inelastic contributions, Raman scattering, FDOM, and chlorophyll fluorescence are included.The subscript t=R+Y+C+E, where E is elastic scattering.The viewing azimuth angle is zero in Fig. 4, i.e., the solar principal plane is used.Figure 4(a) shows that the inelastic fraction for [Chla] surf =0.03 mg m −3 has little angular dependence up to the viewing zenith angle of 50 degrees.Then the fractions start to drop off.The fractions at 443 nm and 490 nm show smaller angular variations than longer wavelengths at 555 nm. Figure 4(b), on the other hand, shows a larger angular variation for all wavelengths.Note that the angular variation of radiance due to inelastically scattered light should be weak.The angular variation in the inelastic fraction shown here is mostly due to the angular variation of total radiance field L t, 0 − , which is highly dependent on the phase function, and single scattering albedo for the scattering medium.We have also confirmed that the azimuth viewing angle dependence of the inelastic fraction is very weak (plot not shown).The overall angular dependence of the inelastic scattering is smooth and limited as indicated by Fig. 4. Figure 5 shows the FDOM and Raman inelastic contribution fraction as a function of detector depth in water for three wavelengths.In Fig. 5, subscript u indicates the radiance is for nadirviewing.For a small value of [Chla] surf =0.03 mg m −3 , the FDOM fraction 100 • L Y,u /L t,u has small changes as the detector depth is larger, for wavelengths shorter than 490 nm.The fractions, however, increase rapidly for the wavelength of 555 nm, with the maximum fraction reaching 18 % around 70 meters.The Raman scattering fraction is larger than FDOM, but with similar trends as a function of depth and wavelength.Raman scattering contributes near 70% for λ=555 nm at 70 meters.For a large value of [Chla] surf =1 mg m −3 , the Raman scattering fraction decreases for all wavelengths as the detector goes deeper.The FDOM fraction for 490 nm and 555 nm also decreases for [Chla] surf =1 mg m −3 .This is not the case for λ=443 nm, however, which decreases first and then increases.This non-monotonic behavior exists because the rate of fractional change depends on how the absolute radiance contribution changes for all three components: elastic scattering, Raman scattering, and fluorescence.Different rates of change for the three components lead to different fractions at different detector depths.
Figure 6 shows the chlorophyll fluorescence fraction at 685 nm as a function of detector depth for five [Chla] surf values.For [Chla] surf values smaller than or equal to 0.1 mg m −3 , the chlorophyll fluorescence increases monotonically as detectors go deeper.For [Chla] surf values larger than 0.1 mg m −3 , the chlorophyll fluorescence fractions first increase and then decrease after they reach peak at depth.The reason is that a larger [Chla] surf value leads to more fluorescence at shallow water depths.However, a larger [Chla] surf value also causes larger extinction coefficients at excitation wavelengths.As a consequence, the excitation radiation attenuates faster, which induces less fluorescence at deeper water depths.It is also observed from Fig. 6 that the fluorescence fractions change at different rates at different depths.This is because radiances from these different sources, elastic, Raman, and fluorescence monotonically decrease at different rates.The different rates of change for these radiances have led to the chlorophyll fluorescence fraction changes at different rates.

Polarized features
A special feature of this work is that the fluorescence is built in a fully polarized radiative transfer model.Thus we can study the impacts of the inelastic scattering to the polarized radiation field.The Degree of Linear Polarization (DoLP) is an important parameter to study polarized radiation field: Figure 7 shows DoLP at 685 nm as a function of viewing angles for two different detector locations.Figures 7(a Figure 7 shows that the angular distribution of DoLP is similar for all the cases, with a minimum value of near zero centered at the backscattering direction of solar light direction refracted by the water surface.The maximum values vary and are distributed in an arched area which has the scattering angle of 90 • , a feature determined by the single scattering reduced Mueller matrix [65].For elastic scattering, the peak value of DoLP decreases as detector depth increases.For DoLP including both inelastic and elastic scattering, the angular pattern is similar.However, DoLP is greatly reduced due to inelastic scattering as fluorescence is completely    The impacts of inelastic scattering to the orientation of the polarization ellipse (OPE) is studied in Fig. 9.In this work the OPE is defined as: which means that we do not adjust the angle to be within 0 and 180 • in order to avoid numerical instability as Q could be zero.The layout of Fig. 9 is the same as Fig.

Conclusion
A full-physical radiative transfer model has been developed for coupled atmosphere and ocean systems, which treats polarization exactly and includes both elastic and inelastic scattering.
In inelastic scattering, Raman scattering, FDOM and chlorophyll fluorescence are included.Sensitivity studies were performed to evaluate the impacts of inelastic scattering on the polarized radiation field in case 1 ocean waters.Chlorophyll fluorescence was shown to have negligible contributions to the total radiation field in the visible, except in a narrow band center at 685 nm.
For relatively clear waters ([Chla] surf <1 mg m −3 ), Raman scattering was shown to have a larger contribution to the total radiation field than the FDOM.This comparison of Raman scattering and FDOM signals is however based on the bio-optical model used for Case 1 waters.The result for Case 2 waters are likely to be different, which is of interest for a future study.The FDOM fraction was shown to be small at the two spectral ends of 412 nm and 690 nm, and larger in the middle with peak around 555 nm, which is determined by the emission function of [49] used in this work.The angular distribution of the inelastic scattering fraction was found to be smooth and limited.At different depths, the fractions of different inelastic scattering sources varied in a large range, from 0 to nearly 100 %, depending on the depth, wavelength, and water turbidity.The inelastic scattering contributions were found to have apparent impacts on the underwater DoLP.These effects are most significant at 685 nm where chlorophyll fluorescence peak is located.Inelastic scattering played a smaller role to OPE angular distribution.However, for a detector at depth, the OPE difference between elastic and inelastic scattering was noticeable.This reflects the complicate compound effect of the inelastic scattering to the polarized radiation field.This work is critical for understanding the underwater polarized radiation field impacted by a set of interacting scattering sources, and can ultimately be used to develop new remote sensing algorithms for ocean color based on polarimetry.

Fig. 1 .
Fig. 1.Single scattering albedo of ocean waters for a set of [Chla] values

Fig. 3 .
Fig. 3. Absolute and relative inelastic scattering contribution to a detector just below the ocean surface for a set of [Chla] surf values.The Y-axis labels denote the different quantities plotted.The subscript R, Y, and C denotes Raman, FDOM, and chlorophyll fluorescence, respectively.The legend in Fig. 3(a) applies to all figures in Fig. 3.Note the different wavelength range in Figs.3(e) and 3(f).

Fig. 4 .
Fig. 4. Fraction of inelastic scattering just below the ocean surface as a function of viewing zenith angle in the principal plane.

3 Fig. 5 .
Fig. 5. Fraction of CDOM fluorescence and Raman scattering at nadir as a function of detector depth.

3 Fig. 6 .
Fig.6.Fraction of chlorophyll a fluorescence at 685 nm at nadir as a function of detector depth.
Figure 7 shows DoLP at 685 nm as a function of viewing angles for two different detector locations.Figures 7(a) and 7(c) are the DoLP originating from elastic scattering only; and Figs.7(b) and 7(d) are the DoLP originating from all the sources, including both elastic scattering and inelastic scattering.The colorbar in Fig.7shows the value of DoLP.The semicircles show the viewing angle plotted in polar coordinate.For an arbitrary pixel in the semicircle, the distance between the pixel to the origin (center of the semicircle) is the viewing zenith angle; the angle between the positive x-axis (from the origin to the right) and the line segment connecting the pixel and the origin is the viewing azimuth angle.The values of the viewing azimuth angles are shown by the bold numbers around the semicircle; while the viewing zenith angle is shown by the numbers along the positive x-axis.Figure7shows that the angular distribution of DoLP is similar for all the cases, with a minimum value of near zero centered at the backscattering direction of solar light direction refracted by the water surface.The maximum values vary and are distributed in an arched area which has the scattering angle of 90 • , a feature determined by the single scattering reduced Mueller matrix[65].For elastic scattering, the peak value of DoLP decreases as detector depth increases.For DoLP including both inelastic and elastic scattering, the angular pattern is similar.However, DoLP is greatly reduced due to inelastic scattering as fluorescence is completely

Fig. 7 .
Fig. 7. Angular distribution of DoLP at 685 nm for elastic scattering and total (elastic + inelastic) scattering at two different detector locations.[Chla] surf =0.1 mg m −3 .(a) Elastic scattering for a detector just below ocean surface.(b) Total scattering for a detector just below the ocean surface.(c) Elastic scattering for a detector 30 meters below the ocean surface.(d) Total scattering for a detector 30 meters below the ocean surface.

Fig. 8 .
Fig. 8. DoLP at 685 nm as a function of viewing zenith angle for three azimuth plane at two different detector locations.[Chla] surf =0.1 mg m −3 .(a) Detector just below ocean surface.(b) Detector 30 m below ocean surface 7, except that the OPE is plotted.If only elastic scattering is included, the OPE just below the ocean surface shows a positive region near backscattering direction of the solar incident light.It then transits from positive to negative around zenith angles between 20 • to 40 • , indicating a sign change of U/Q.For the detector 30 m below the surface, the positive region is confined in a smaller cone, approximately within 20 • of the viewing zenith angle.When inelastic scattering is taken into account, the OPE near the surface shows a similar pattern to the elastic figure.However, the OPE angular pattern at 30 m below the surface is different from the one with elastic scattering only.The main difference is the angular range of the positive values of OPE.The subtle difference of OPE between elastic scattering and total scattering fields reflects the interaction between elastic scattering and less polarized inelastic scattering, which demonstrates the value of this work.