1. Introduction

An accurate description of radiative transfer in a coupled system consisting of two media with different refractive indices is important in a variety of different applications. For optical satellite remote sensing of water bodies [1–4] and for interpretation of images of biological tissue [5–7] an accurate radiative transfer model is an indispensable tool [8]. For geometries other than the plane-parallel or slab geometry, for media with irregular boundaries or interfaces, for problems involving polarization or complicated volume scattering functions (VSFs), Monte Carlo methods are useful. The Monte Carlo approach involves the use of probabilistic concepts to simulate the trajectories of individual photons. The discrete ordinate radiative transfer (DISORT) method [9, 10], invariant embedding [11], and the multicomponent approximation [12] are alternative methods for solving the radiative transfer equation. All these methods are computationally faster than the Monte Carlo method but are not as easily adjustable to new problems of interest. Mobley et al. [13] showed that underwater scalar light fields computed by several different methods, including five different Monte Carlo methods, give similar results for a limited set of test cases, and Gjerstad et al. [14] made a detailed comparison of scalar light fields computed by coupled atmosphere-ocean DISORT and Monte Carlo methods, finding excellent agreement between results simulated by the two techniques. Further, Hestenes et al. [15] obtained excellent agreement when comparing scalar light fields computed by a coupled air-tissue DISORT method with those computed by a Monte Carlo method.

Recent works on polarized radiative transfer in the atmosphere include studies of 3D effects of cirrus clouds [16] as well as impact of aerosols [17], while previous Monte Carlo models for polarized radiative transfer in a coupled atmosphere-ocean system include those of Kattawar and Adams [18] and Tynes et al. [19]. Kattawar and Adams [18] analyzed the effect of the refractive index on the radiance and polarization. Tynes et al. [19] further developed the code of Kattawar and Adams [18] to include a sophisticated time saving Monte Carlo estimation techique. In addition, Tynes et al. [19] calculated the complete Mueller scattering matrix, which contains all scattering information about the two media. These models did not include realistic Mueller matrices for hydrosols and therefore could not be used to analyze the effect of the particulate oceanic composition on the radiance and polarization of the underwater and backscattered light field.

Ocean water is often categorized as case 1 or case 2 [20]. Case 1 water, which is found in open oceans, has optical properties that depend only on chl-a. Retrieval of the chl-a concentration in such waters from remote sensing data is usually based on empirical relations, such as C = A[R(445 nm)/R(550 mn)]^B, where R(445 nm) and R(550 nm) are retrieved reflectances at 445 nm and 550 nm, respectively, C is the chl-a concentration, and A and B are coefficients obtained from a regression analysis [20]. In case 1 waters this relation gives reasonably good agreement with the chl-a concentration found in the upper layers of the ocean.

In coastal waters, which are categorized as case 2, the optical properties are influenced also by inorganic particles and yellow substance [20]. Inorganic particles are mainly found in waters affected by river discharge, and they will influence the measured light field, making it impossible to apply commonly used algorithms and inversion procedures for case 1 waters to obtain the chl-a concentration [20]. Therefore, in order to retrieve the chl-a concentration in case 2 waters additional measurements must be performed.

Here we present simulations suggesting that it may be possible to determine the ratio between the amounts of organic and inorganic particles from measurements of the full Stokes vector of the underwater or backscattered light field. The foundation for this suggestion is that the real part of the refractive index for organic particles is different from that for inorganic particles.

We used the T-matrix code of Mischenko and Travis [21] to compute Mueller matrices for spherical as well as spheroidal organic and inorganic particles. For each type of particles we used Junge size distributions with different values of the exponent γ as well as different asphericities and refractive indices. Among these three parameters, within their expected ranges, we found the real part of the refractive index to most significantly affect the Mueller matrix.

In order to examine whether the differences between the light fields produced by organic and inorganic particles (with different Mueller matrices) would be measureable in coastal waters, we developed a Monte Carlo code that simulates polarized radiative transfer in a coupled atmosphere-ocean system. Using this code, which is a further development of a scalar Monte Carlo code developed by Gjerstad et al. [14], we simulated the transport of photon packets, each containing information about the full Stokes vector. We have compared results obtained with this Coupled Atmosphere-Ocean Polarized Monte Carlo (CAO-PMC) code with tabulated results obtained with other Monte Carlo codes [19], and found good agreement. The accuracy of the CAO-PMC code was also investigated in Sommersten et al. [22], where the results were compared to those obtained from a vectorized version of the DISORT method.

2. Methods

2.1. Mueller matrix

For scattering by air molecules or density fluctuations in the water we use the Mueller matrix for Rayleigh scattering given by [23]

where Θ is the scattering angle and ρ is the depolarization factor. The depolarization factor depends on the anisotropy of the molecules causing the scattering. In the current study the anisotropy is neglected and the depolarization factor is set to zero.

To find the Mueller matrices for scattering by particles we used the T-matrix code developed by Mischenko and Travis [21]. This code, which is available from the NASA Goddard institute for space studies [24], can be used to compute light scattering by polydisperse, randomly oriented, rotationally symmetric particles.

In the literature we found typical values for the refractive index (relative to water) for organic and inorganic particles to be about 1.05+i 0.005 and 1.15+i 0.001, respectively. These values are representative for those found by flow cytometric techniques [25] and by other means [26–29]. The reported refractive-index values vary to a certain degree, but there is always a significant difference between the two groups of particles, mainly due to the higher percentage of water in organic particles compared to that in inorganic particles [27].

Figures 1 and 2 show Mueller matrix elements for scattering by organic and inorganic spherical particles and spheroidal particles of asphericity 2, computed by means of the T-matrix code of Mischenko and Travis [21]. Because the particles are assumed to be randomly oriented and possess certain symmetry properties, 8 of the 10 Mueller matrix elements not present in Figs. 1 and 2 are negligible, whereas M ₂₁ = M ₁₂ and M ₄₃ = -M ₃₄ [30].

Figure 1 shows Mueller matrix elements for a Junge size distribution of spherical particles with an exponent of γ = -3, a lower cut-off radius r ₁ of 0.1 μm, an upper cut-off radius r ₂ of 5 μm, and an effective radius of 1.28 μm. Ideally we would have used a larger cut off radius for the size distributions, but found it difficult to obtain convergence. By fine tuning the parameters in the T-matrix code we could probably have obtained convergence with a slightly larger cut off radius. However, the Mueller matrices are in agreement with the laboratoy measurements of Volten et al. [29], as pointed out at the end of this section. The number of Gaussian quadrature points used in the size averaging was 44. The accuracy of the T-matrix computations was set to 0.001, and a wavelength in water of 488 nm was used (corresponding to a wavelength of 653 in air for a real part of the refractive index in water of 1.34 relative to that in air).

By comparing the M ₁₁ elements in Fig. 1 for organic and inorganic particles, we see that the latter have much larger backscattering as well as a scattering enhancement close to 180°. Also, it is of interest to note that while the M ₁₂ elements in Fig. 1 are very similar for Rayleigh scattering and scattering by inorganic particles, the M ₁₁ elements differ considerably, implying that the scattering phase function can be used to distinguish between these two kinds of scattering. Further, we see that the M ₁₂ element for organic particles has a much larger absolute value at a scattering angle of 90° than the M ₁₂ element for inorganic particles. There are large differences also for the other elements, exept for the M ₂₂ element, which is equal to 1 for spherical particles.

 

Fig. 1. Mueller matrix elements for spheres compared to those for Rayleigh scattering.

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Fig. 2. Mueller matrix elements for randomly oriented spheroidal particles with an asphericity of 2 compared to those for Rayleigh scattering.

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For the spheroidal particles in Fig. 2 we see similar features as those described above for spherical particles. Note, however, that there is no backscattering enhancement of the M ₁₁ element for inorganic particles close to 180°. The difference between the M ₁₂ elements for organic and inorganic particles is even more pronounced than in Fig. 1. Also, compared to Fig. 1, the curves in Fig. 2 for spheroidal particles are generally smoother mainly because the spheroidal particles are randomly oriented while the spherical particles in effect are uniformly oriented and show ripples in the angular scattering distribution due to resonances. The M ₂₂ element in Fig. 2 is seen to deviate from its value of 1 for spherical particles in Fig. 1. The M ₂₂, M ₃₃, and M ₄₄ elements in Fig. 2 are much closer to the empirical values found in natural waters by Voss and Fry [31, 32] than the corresponding elements in Fig. 1.

The choice of an asphericity of 2 for the computations in Fig. 2 was somewhat arbitrary, the philosophy being that any non-spherical shape represents the shape of a natural particle better than the spherical shape. As mentioned above, this conclusion also follows from comparisons of our computed results to those for Mueller matrices found in natural waters. However, we have carried out computations for different shapes and concluded that the real part of the refractive index of scattering particles has a much larger impact on the Mueller matrix than their shape. Our main objective is not to provide precise results for the Mueller matrix or the underwater and backscattered light fields asssociated with a particular size/shape distribution of particles, but to obtain an approximate quantification of the role of the particluate composition in natural waters.

The M ₁₂ element of the Mueller matrix is of particular interest in our context. From Fig. 2 it follows that M ₁₂ depends strongly on the real part of the refractive index and that initially unpolarized light will be polarized to a degree of about 0.9 after scattering at 90° by organic particles. On the other hand, unpolarized light scattered by inorganic particles at 90° will remain almost completely unpolarized. This result is consistent with the experimental findings of Volten et al. [29], according to which the M ₁₂ element for silt particles has a maximum negative value of about -0.38 for small particles and -0.25 for large particles. Algae on the other hand, were found to have maximum negative values ranging from -0.5 to -0.85. Also, the angular position of the maximum negative value of the M ₁₂ element depends on the refractive index. For algae it is close to 90°, whereas it is close to 100° for inorganic particles. Figure 1 (spherical particles) and Fig. 2 (spheroidal particles) show that the M ₃₄ element for mineral particles is non-zero in the angular range from approximately 60° to 180°, implying that mineral particles can convert linearly polarized light into circularly polarized light.

2.2. Monte Carlo simulations

Consider a one-dimensional geometry in which both the atmosphere and the ocean consist of horizontal plane-parallel layers. We study polarized radiative transfer in this system by using Monte Carlo simulations to determine the random walk of photons incident at the top of the atmosphere until they are absorbed or scattered back to space. In addition to absorption and Rayleigh scattering [33], our simulations include scattering by particulate matter in the ocean of organic and inorganic origin. The scattering processes included in our CAO-PMC model are described in terms of tabulated Mueller matrices.

Each of the horizontal layers in the system is described by its own Inherent Optical Properties (IOPs). These are the refractive index n, the absorption coefficient a, and the scattering coefficients b_r, b_i, and b_o for Rayleigh scattering, scattering by inorganic particles, and scattering by organic particles, respectively. The total scattering coefficient b is the sum of the three separate scattering coefficients, i.e. b = b_r +b_i+b_o.

The CAO-PMC code traces the path of each photon through the system layer by layer. Each photon is assumed to start in a given direction at a given point at the top of the atmosphere.

When a photon passed through one of the predefined horizontal surfaces z = constant, it was counted, and the current radiance value I(z, θ, ϕ) at that level in the direction (θ, ϕ) was then augmented by 1/cosθ. At the same time, the current value of Q(z,θ, ϕ) was augmented by Q/cosθ, the current value of U(z,θ, ϕ) was augmented by U/cosθ, and the current value of V(z,θ,ϕ) was augmented by V/cosθ. The radiance value I of an incident photon was always set equal to 1. The full range 2π of the azimuth angle ϕ was divided in 40 equal intervals and the nearly full range of the cosine of the polar angle μ = cosθ was divided in 58 equal intervals plus one smaller interval close to μ = 1 and another close to μ = -1 to produce polar caps subtending the same solid angle as each of the quads, as described by Mobley [34]. The albedo of the ocean bottom was assumed to be 0 so that all radiation reaching the bottom was absorbed.

 

Fig. 3. Radiance I (number of photons per solid angle divided by the number of photons incident at the top of the atmosphere), degree of linear polarization $P_{\ell }=\sqrt{Q^2+U^2}$, and degree of circular polarization P_c = V/I for downwelling light at the bottom of the ocean, computed by our CAO-PMC code, versus polar angle θ for azimuth angles ϕ of 0°, 90°, or 180°, as indicated by the color code. Unpolarized sunlight was incident at a polar angle θ of 120° and an azimuth angle of 0°. There was 5% Rayleigh scattering and 95% scattering either by organic particles (org) or inorganic particles (inorg). The total optical depth was τ = 1.15, of which τ _atm = 0.15 and τ _water = 1.00, and the wind speed was 0 m/s. The absorption was zero both in the atmosphere and the ocean.

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One advantage of using Monte Carlo simulations is that it is relatively easy to include effects of surface waves. When the surface of the ocean is roughened by wind, it is necessary to find a probability function for the normal unit vector n̂ of a surface facet. This was determined from the normalized probability density function given by Cox and Munk [35]. When the wind speed is small, one can neglect multiple reflections from neighboring facets and shadowing effects.

3. Results and discussion

We considered the oceanic scattering to be either a mixture of Rayleigh scattering and scattering by organic particles or a mixture of Rayleigh scattering and scattering by inorganic particles. For simplicity, we let the atmosphere remain non-absorbing and purely Rayleigh scattering. First, we considered a situation in which the oceanic scattering was 95% due to scattering from organic particles and 5% due to Rayleigh scattering. When a scattering event occurred, we picked a random number ρ in the interval [0,1]. If it was less than 0.05, we used the Mueller matrix for Rayleigh scattering, and otherwise we used the Mueller matrix for scattering by organic particles. Second, we considered precisely the same situation as just described, except that we replaced the organic particles by inorganic particles.

In Fig. 3 we compare the radiance I, the degree of linear polarization $P_{\ell }=\sqrt{Q^2+U^2}$/I, and the degree of circular polarization P_c = V/I for downwelling light in water contaning only organic particles to those in water containing only inorganic particles. Comparing the radiance curves for ϕ = 0° in the left panel of Fig. 3, we see that both have a maximum value at a polar angle of about θ _max = 140°, which is the polar angle of the direct radiance in the water. The radiance at this polar angle consists of both scattered and direct light. The high radiance values at polar angles close to θ _max are caused by the forward lobe of the VSF (the M ₁₁ element of the Mueller matrix). For polar angles in a small region around θ _max, the radiance values in water containing organic particles are higher than in water containing inorganic particles, because organic particles scatter more strongly in the forward direction. For polar angles outside a small region around θ _max the radiance values are seen to be higher in water containing inorganic particles than in water containing organic particles, as expected, since inorganic particles have a larger refractive index than organic particles and therefore scatter more in intermediate and backward directions.

For ϕ = 90° and for ϕ = 180° the radiance values are rapidly increasing at a polar angle of about 132°. This effect is due to the fact that the diffuse downward radiance in the atmosphere is refracted into a cone of light in the water limited by the critical angle.

From the middle panel of Fig. 3 we see that, except for observation directions with polar angles close to θ _max = 140°, the degree of linear polarization, given by P_ℓ = (Q ² + U ²)^1/2, is considerably higher in water containing organic particles than in water containing inorganic particles. This result is expected because the absolute value of M ₁₂ is much larger for organic particles than for inorganic particles (see Fig. 2). The largest ratio between P _ℓ for scattering by organic particles and P _ℓ for scattering by inorganic particles is found to occur in the direction θ = 103°, ϕ = 0°. In this direction the ratio is 12.8, which is a very large value, but the value for the degree of linear polarization associated with inorganic particles is so small that it may be difficult to measure it in situ. At θ = ϕ = 90° the degree of linear polarization is found to be 1.9 times higher for scattering by organic particles than for scattering by inorganic particles. Here the two values for the degree of linear polarization are higher, implying that it may be easier to measure them in situ.

From he right panel of Fig. 3 we see that at θ = 110°, ϕ = 90° the degree of circular polarization for scattering by organic particles is about 2.5 times larger than for scattering by inorganic particles. Also, we see that the degree of circular polarization is larger for scattering by organic particles than for scattering by inorganic particles. The explanation of this is (i) that downwelling circularly polarized light is obtained when upwelling linearly polarized light is created through backscattering by particles in the ocean and then is totally reflected at the water-air interface, and (ii) that the degree of linear polarization obtained for scattering by organic particles is significantly larger than that obtained for scattering by inorganic particles because the magnitude of the M ₁₂ element of the Mueller matrix is larger for scattering by organic particles (see Fig. 2).

To determine how the results in Fig. 3 would be affected by a wind-ruffled ocean surface, we used a wind speed of 5 m/s. Figure 4 shows results corresponding to those in Fig. 3 but for a wind-ruffled ocean surface. The major difference between the results obtained for a wind-ruffled ocean surface compared to those obtained in the absence of wind-ruffling, is that the curves are smoother in the presence of wind-ruffling. By comparing the curves for ϕ = 0° in the left panels of Figs. 3 and 4, we see that this smoothening effect pertains most prominently to a region around the peak of the radiance at about θ _max = 140°. We also clearly see the smoothening effect by comparing the curves for ϕ = 180° in the middle panels of Figs. 3 and 4 in a region around the critical angle of about 48°, corresponding to a polar angle of observation of 132°, above which the conversion from linear polarization to circular polarization takes place. The maximum values for the degree of linear polarization are seen to be about the same with and without wind-ruffling, while the degree of circular polarization is seen to decrease with wind-ruffling.

In the next simulation we ignored wind-ruffling, let the atmosphere remain non-absorbing and purely Rayleigh scattering, but introduced two changes in the optical properties of the ocean. The optical thickness of the water was assumed to be 4, implying that the bottom of the ocean was at an optical depth of τ = 4.15. We also introduced absorption in the ocean, so that 25% of the oceanic attenuation was due to absorption while 75% was due to scattering. Figure 5 shows the radiance I, the degree of linear polarization P _ℓ, and the degree of circular polarization P_c at the bottom of the ocean computed for this case. We see that the radiance and the degree of circular polarization are about ten times smaller than in Figs. 3 and 4, while the degree of linear polarization is about the same as in Figs. 3 and 4.

 

Fig. 4. Same as Fig. 3 except that the wind speed is 5 m/s.

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In the direction of the direct beam at θ = 140°, ϕ = 0° the radiance in the left panel in Fig. 5 is seen to be larger for organic particles than for inorganic particles even though the attenuation coefficient c is the same in both cases, implying that the direct beams should be equally much attenuated. The explanation is that light is not only scattered out of the direction of the direct beam, but also scattered into this direction through multiple scattering. In the direction of the direct beam the ratio of the radiance obtained for organic particles to that obtained for inorganic particles is 2.5, which is larger than in the cases of optically thinner water considered previously, the reason being that there is more multiple scattering in the optically thicker medium and because more light is scattered into the direction of the direct beam by organic particles (strong forward scattering) than by inorganic particles.

The radiances in the left panel of Fig. 5 for ϕ = 90° and ϕ = 180° are seen to be nearly steadily increasing in the whole range from θ = 90° to θ = 180°, implying that the radiance has not become isotropic deep in the water column. On the contrary, it has higher values in directions straight downward (θ = 180°) than in directions close to the horizontal (θ = 90°). Also the radiance values for ϕ =0° are seen to be higher around θ =180° than around θ =90°, indicating again that the radiance is stronger in directions straight downward than in directions close to the horizontal.

Figures 3–5 show graphs for each of the three observables I, P_ℓ, and P_c as a function of the polar angle θ in the range [90°,180°] at one particular depth z for three different azimuth angles ϕ = 0°, ϕ = 90°, and ϕ = 180°. Since each of these three observables is a function of three variables, i.e. depth z, polar angle θ , and azimuth angle ϕ , it would take a lot of space to present graphs for a variety of depths and azimuth angles. However, by presenting the computed results in the form of contour plots, we can display all of them in a compact manner that provides a good overview. Figures 6 and 7, which pertain to the same physical situation as in Fig. 5, show contour plots for τ = 0.075 (middle of the atmosphere, first column), τ = 0.15⁺ (just above the ocean surface, second column), τ = 0.15^- (just below the ocean surface, third column), and τ = 3.03 (deep in the ocean, fourth column). At the top of the atmosphere (τ = 0, not shown) there is no diffuse downwelling light, implying that the downwelling light is the direct, unpolarized solar beam.

 

Fig. 5. Radiance I, degree of linear polarization $P_{\ell }=\sqrt{Q^2+U^2}$, and degree of circular polarization P_c = V/I for downwelling light at the bottom of the ocean, computed by our CAO-PMC code, versus polar angle θ for azimuth angles ϕ of 0°, 90°, or 180°, as indicated by the color code. Unpolarized sunlight was incident at a polar angle θ of 120° and an azimuth angle of 0°. There was 5% Rayleigh scattering and 95% scattering either by organic particles (org) or inorganic particles (inorg). The total optical depth was τ = 4.15, of which τ _atm = 0.15 and τ _water = 4.00, and the wind speed was 0 m/s. The absorption was zero in the atmosphere. The attenuation in the ocean was 75% due to scattering and 25% due to absorption.

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In the middle atmosphere. In the middle of the atmosphere at τ = 0.075 (first column of Figs. 6 and 7) the attenuated direct incident radiance at θ = 120° and the attenuated specularly reflected radiance at θ = 60° dominate, regardless of whether the scattering in the ocean is due to organic or inorganic particles. In addition there is diffuse light, which is strong in all azimuth directions ϕ for polar angles close to θ = 90°. The degree of linear polarization (second row) is highest when the direction of the scattered light is at 90° to the direction of the incident light, i.e. for a scattering angle of Θ = 90°. The curved solid line in the second row of the first column in each of Figs. 6 and 7 shows all combinations of θ and ϕ that correspond to a scattering angle of Θ = 90°. This curved line follows precisely the maximum values of the calculated degree of linear polarization. This result is expected since the influence of scattering by particles in the ocean is small and since the atmosphere is assumed to have only Rayleigh scattering, which gives completely linearly polarized scattering in directions at right angles to the incident beam [36].

Immediately above the ocean surface. When the scattering in the ocean is due to inorganic particles (second column, second row of Fig. 6), the upwelling light at θ = 40°,ϕ = 0° just above the ocean surface (τ = 0.15⁺) has a smaller degree of linear polarization than that for organic particles (Fig. 7). This difference is probably caused by the fact that the magnitude of the M ₁₂ element of the Mueller matrix is significantly larger for scattering by organic particles than for scattering by inorganic particles, as shown in Fig. 2. Therefore, it should be possible to do measurements just above the ocean surface to infer information about the fraction of organic vs. inorganic particles in the water column. Finally, we note that there is virtually no circularly polarized component of the light just above the ocean surface (τ =0.15+), regardless of whether the ocean contains organic or inorganic particles (second columns, third rows in Figs. 6 and 7). As explained above, the lack of circularly polarized light is due to the fact that the atmosphere is assumed to have only Rayleigh scattering, for which the V component of the Stokes vector is zero.

 

Fig. 6. The panels in the first row show radiances (the numbers on the colorbar to the right correspond to the exponent of 10 on the vertical axis in Figs. 3–5, e.g. 0.5 implies a radiance value of 10^0.5), those in the second row show degrees of linear polarization, and those in the third row show degrees of circular polarization for the case in which the water contains inorganic particles. The four columns pertain to different optical depths τ, as indicated by the column header. Below the ocean surface, in the third and the fourth columns, the critical angle is marked with a solid line. The column for τ = 0.15⁺ shows results immediately above the ocean surface, while the column for τ = 0.15^- shows results immediately below the ocean surface.

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Immediately below the ocean surface. Just below the ocean surface at τ = 0.15^- (third columns in Figs. 6 and 7), we see that the upwelling radiance (first row) is higher in water containing inorganic particles than in water containing organic particles. The downwelling radiance distributed over the entire hemisphere in the atmosphere is confined to an angular cone in the water limited by the critical angle, causing a discontinuity in the derivative of the radiance across the polar angle of observation equal to the critical angle. The upwelling radiance in the ocean is reflected at the water-air interface causing the radiances in the first row and third column in each of Figs. 6 and 7 to be approximately symmetric about θ = 90°, but only in the proximity of this angle, which is marked with a broken line in the two upper parts of the third column in Figs. 6 and 7. In other words, for upwelling light in the ocean in directions close to the horizontal direction (θ = 90°) the ocean surface acts as a perfect mirror making the downwelling light become a reflection of the upwelling light.

 

Fig. 7. The same as Fig. 6 exept that the particles are organic.

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The linearly polarized component of the light (second row, third column) exhibits the same symmetry as the radiance. The degree of linear polarization for the downwelling light is slightly lower than the degree of linear polarization for the upwelling light because some of the linearly polarized upwelling light is converted into circularly polarized light upon total reflection at the water-air interface. The degree of circular polarization of the downwelling light is therefore high just below the ocean surface. It is higher in water containing organic particles than in water containing inorganic particles because the generation of circularly polarized light upon total reflection depends on the degree of linear polarization of the incident upwelling light, which is larger for scattering by organic particles than for scattering by inorganic particles because the magnitude of the M ₁₂ element of the Mueller matrix is larger for scattering by organic particles (see Fig. 2). However, the degree of circular polarization for the upwelling light is higher in water containing inorganic particles than in water containing organic particles. This difference, which is not visble in the contour plots textcolorredon paper but barely visible when displayed on a computer screen. may be due to the nonzero M ₃₄ element of the Mueller matrix for inorganic particles. This element governs the conversion of linearly polarized incident light into circularly polarized reflected light (see Fig. 2).

Deep in the ocean. Deep in the ocean at τ = 3.03 (fourth columns in Figs. 6 and 7), we see that the radiation is still concentrated in directions around the polar angle θ _max = 140° of the direct light and around azimuth angles of ϕ = 0° and ϕ = 180°. Also, the upwelling radiance is very small, but, as expected, it is higher in water containing inorganic particles than in water containing organic particles. The abrupt behaviour of the radiance across the polar angle of observation equal to the critical angle is no longer seen. The maximum degree of linear polarization is about twice as high in water containing organic particles as in water containing inorganic particles. The degree of circular polarization is small.

4. Conclusions

Our results show that the particulate composition of oceanic water governs the degrees of linear and circular polarization of the underwater light field. Both the degree of linear polarization and the degree of circular polarization are higher in water containing organic particles than in water containing inorganic particles. This conclusion holds for the two different optical thicknesses of the water (τ = 1 and τ = 4), which we have investigated and also for the case of a wind-ruffled ocean surface. The aim of this work was to investigate wether it be possible to determine the fraction of organic vs. inorganic particles from measurements of the full Stokes vector of the underwater light field. For the set of test cases presented in this paper we have demonstrated that this is possible.

However, Mueller matrices have been measured for a large number of different phytoplankton species [29], but only for a limited number of water samples containing inorganic particles. As a continuation of this work, it would be of interest to determine wether the effects we have demonstrated hold for a broader range of naturally occurring particles in water. It would also be of interest to investigate the effects of a realistic amount of aerosols in the atmosphere and how much a variation of the amount and type of aerosols would affect the polarization measurements discussed in this paper. Further, it would be desirable to include spectral information in future simulations and measurements.

From the discussion in section 3 it is clear that the polarized light field varies significantly with the fraction of organic vs. inorganic particles in the water. This conclusion is most evident for the light field in the water but is also valid for the light field above the water surface. Thus, an instrument for measuring the optical properties displayed in Figs. 6 and 7 could be used to estimate the fraction of organic vs. inorganic particles in the water. Such an instrument should be designed to measure simultaneously in many different directions, and its orientation when lowered into the water would follow from knowledge of the solar angle and the measurements in Figs. 6 and 7.

The estimation of the fraction of organic vs. inorganic particles in the manner discussed in this paper, would rely mainly on relative measurements, making the measurement procedure robust. The measurements most likely to give valuable information about the fraction of organic vs. inorganic particles are (i) the degree of linear polarization in the water column as well as above the ocean surface, (ii) the degree of circular polarization of downwelling light in the water column, and (iii) the degree of circular polarization of upwelling light in the water column. Items (i) and (ii) are related to differences between the M ₁₂ Mueller matrix elements for organic and inorganic particles, whereas item (iii) is related to the differences between the M ₃₄ Mueller matrix elements.

It is outside the scope of this paper to present a method to derive particle composition from polarization measurements, but some thoughts on this subject are in order. The simulations presented in this paper encompass Rayleigh scattering in the atmosphere and Rayleigh scattering plus scattering by either organic or inorganic particles in the water. As mentioned above, a natural next step would be to include the effects of aerosols in the atmosphere, but it would be equally important to consider the effects of a mixture of organic and inorganic particles in the water.

To solve the inverse problem of retrieving the fraction of organic vs. inorganic particles it will be necessary to solve many forward polarized radiative transfer problems with different values of the retrieval parameters, which represent the input to the solution of each forward problem. In addition to the fraction of organic vs. inorganic particles in the water the retrieval parameters would include parameterizations of water particle shape, aerosol optical depth and fraction of large vs. small aerosols (in the case of a bimodal size distribution) and dissolved organic matter absorption.

To determine the retrieval parameters one could follow a similar procedure as that in described in a recent paper [4] for simultaneous retrieval of aerosol parameters and case 2 water constituents. In that case no polarization information was used, but using forward polarized radiative transfer modeling one can construct a database of synthetic measurements to which the measured data can be compared by means of an optimization scheme in order to determine the retrieval parameters. To construct such a large database, Monte Carlo calculations are too slow so that the use of a deterministic forward radiative transfer model such as that described in [22] would be preferable.