1. Introduction

Spectral radiance (L) is the fundamental radiometric quantity used to describe the radiance field of any medium. It is defined as the radiant power (Φ) in a specified direction (θ, φ) per unit area (A), per unit solid angle (Ω), within a wavelength interval (Δλ), and it is expressed in the units of Wm⁻²sr⁻¹nm⁻¹.

When the radiant flux travels from one medium to the other (air to water or water to air), it undergoes important optical changes at the interface (a layer of separation between the two medium) determined by the radiance transmission coefficient or the radiance transmittance. Transmission of the radiance across the two-medium interface was first geometrically described by Straubel, known as the n² - law for radiances [7–10] which is one of the most significant laws of geometric optics. For further details on the law and the geometry, refer to Appendix A and Fig. 1. This law has been modified for application to the air-sea interface (also, sea-air) by incorporating the loss of radiance through Fresnel reflection [10–13]. The fully emerged radiance from the water after crossing the water-air interface is known as the water-leaving radiance. Thus, the upwelling radiance after transmission through the water-air interface is given by,

 

Fig. 1 Systematic geometrical representation of the transfer of radiance from one medium to another n₁ and n₂ respectively. P is the layer of separation (interface) between the two media, θ₁, θ₂ and θ₃ are the incident, refracted and reflected angles with respect to normal, and θ_c is the critical angle.

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The form of upwelling radiance transmittance [Eq. (3)] derived from the n² - law for radiance exhibits only the geometrical behaviour of radiance transmitted from water to air. Although it is theoretically correct based on its geometrical radiometry [Fig. 1], additional contributions of the water medium to τ_w,a are ignored. Our interest is to quantify these contributions to the upwelling radiance transmittance in natural waters, because of the necessity to determine accurate water-leaving radiances emerging from the water body.

This paper attempts to theoretically improve τ_w,a with two other additional contributions which have been ignored in the previous studies. First, the phenomenon of multiple interactions of the radiance with the water-air interface is incorporated into the existing geometrical upwelling radiance transmittance through the two important parameters namely, the single scattering albedo (ω) and the upwelling average cosines (µ_u). Second, appropriate refractive index values of seawater are determined that replace the conventional use of pure-water refractive index values based on the particulate contribution to the pure seawater. Using the new theoretical formulation obtained from this study, a conversion of sub-surface r_rs to above-surface R_rs is achieved. Finally, the significant improvement on the ocean color quantities L_w and R_rs is demonstrated using in situ data for the entire bio-optical range of natural waters (ω = 0 to 0.97).

2. Theoretical considerations

The geometrical radiometry provides the radiance transmittance only for the ideal case of zero scattering and zero absorption and the multiple interactions of photons within the interface is completely ignored. Optically, the interface is not just a layer of separation between the two media, but many water constituents present within this layer play a crucial role in determining the transmission of radiances through it. The radiance transfer process from water to air is very complex because the water medium (originating side) is denser than the air and is composed of many particles. In this section, factors affecting the upwelling radiance transmittance τ_w,a is addressed based on some theoretical considerations of the water-air interface optical processes.

2.1. Multiple interactions of radiances at the water-air interface

The transfer of upwelling radiance through the water-air interface predominantly relies on the geometrical radiometry, which is however insufficient to explain the interface optical processes considering the fact that multiple interactions of photons occur at the water-air interface and many particles are included in it. Here, two cases are presented: (i) a hypothetical case by assuming the water medium as a non-absorbing medium (i.e., fully scattering), and (ii) a realistic case of natural waters.

2.1.1. For non-absorbing media

Let us designate τ_pw,a as the pure water geometrical upwelling radiance transmittance, the value is simply 0.541 and τ_w,a is the newly derived upwelling radiance transmittance incorporating the multiple interactions of the photons with the water-air interface. The assumption of 0.541 as the transmission factor [Eq. (2)], states that 54.1% of in-water photons escape through the water-air interface leaving the rest of the photons (i.e 45.9%) within the water medium itself. On the first interaction of a photon with the water-air interface, according to the law of conservation of energy, (1-τ_pw,a) is retained within the water column. If there is no absorption within the water-air interface, and if all photons are directed back to the interface again through the upward scattering by the particles (this assumption does not hold naturally), then, the (1-τ_pw,a) photons take a second attempt (i = 2) to cross the interface. Here, i denotes the number of interactions of a single photon with the water-air interface. On the second successive interaction (i = 2), based on the geometry, τ_pw,a part of (1-τ_pw,a) is allowed to cross the interface. As a result, τ_pw,a(1-τ_pw,a) escapes through the interface giving the value of 0.248. For every successive interaction of the photon with the interface, the geometry appears the same and considers it as an new photon entering the interface. At the end of the second successive interaction [τ_pw,a + τ_pw,a(1-τ_pw,a)], 78.9% of photons have crossed the interface. Similarly, on the third successive interaction (i = 3), τ_pw,a(1-τ_pw,a)² escapes the interface producing 90.3% [τ_pw,a + τ_pw,a(1-τ_pw,a) + τ_pw,a(1-τ_pw,a)²] of photons escaping the interface. Finally, for a complete transfer of photons from the water medium to air, it would take at least ten interactions (Table 1) given only if the water medium is totally non-absorbant and all the photons are redirected to the surface.

Table 1. Successive interactions of the photons escaping the water-air interface for a non-absorbing medium.

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On the assumption of above conditions and inclusion of multiple interactions of the photons with the water-air interface on the existing geometry for pure water, the upwelling radiance transmittance is theoretically derived as follows:

According to the law of conservation of energy, for a non-absorbing medium, the maximum value that can be attained is 1. As expected, the new radiance transmittance τ_w,a gives the value of 1. The percentage of photons escaping the interface on each event is comprehensively described in Table. 1, which suggests that the maximum number of interactions can go up to ten (i = 10). Geometrically, on every possible interaction, 54.1% of the photons escapes the interface. Accordingly, on the first interaction (i = 1), 54.1% escapes the interface, 78.9% on the second (i = 2), 90.3% on the third (i = 3), 95.6% on the fourth (i = 4), and 98.0% on the fifth (i = 5), and so on. Note that at the tenth interaction, 100% of the photons would have crossed the interface, and consequently, no more photons are there to undergo for the next interaction to any further extent. From this theory, it can be concluded that there is no more interaction of the upward photons with the water-air interface beyond i = 10 as the number of interactions is always quantized.

2.1.2. For the case of natural waters

The previous section on describing the upwelling radiance transmittance for a non-absorbing medium gives a clear picture to understand the optical processes occurring at the interface. Nonetheless, the assumption made for the non-absorbing case is no longer justified for the case of natural waters. Unlike the non-absorbing case, the photons are subject to get absorbed and scattered depending upon the types of particles (and dissolved substances) present in natural waters. This means that the photons are restricted to further interact with the interface for the successive times. To address the complete multiple interactions of the photons along with the water scattering and absorption processes within the interface, the optical processes must be stated clearly. First, it is essential to have knowledge of how many photons actually survive after a collision with the water molecules and particles. As the water molecules and particles continue to absorb and scatter the photons incident on them, the number of escaping photons depends on the survival index. The survival index is an IOP (inherent optical property) which can be termed as the probability of photon survival or the single scattering albedo $(\omega =b/c)$. The probability of a photon survival describes the fraction of photon scattering ‘b’ over the extinction ‘c’ which gives the percentile estimate of the survival photons after every collision or interaction. The survival index is low for an absorbing medium and high for a scattering medium. In other terms, the survival index of a photon describes the probability of the photon that is scattered rather than absorbed. In an absorbing medium, only less number of photons are gone for further interaction with the interface, whereas in a scattering medium more number of photons are involved. Second, not all survived photons (which are scattered in all directions) travel upward to escape through the interface. Therefore, the probability of upward scattered photons should be estimated in order to determine the proportion of upwelling photons. The proportion of upward photons depends on the angular distribution of the upward scattered flux known as upwelling average cosines, µ_u.

The single scattering albedo multiplied with the average upwelling cosine (i.e., µ_uω) completely describes the percentage of upwelling photons emerging from the water medium after interaction with a water molecule or a particle. The µ_uω plays a role whenever the photon is available for the next successive interactions, therefore, it is multiplied by the successive geometrical upwelling radiance transmittances (τ_pw,a). The radiance transmittance (τ_w,a) for ‘i’ number of interactions with the interface and the molecule/particle is given by

2.2. Particulate contribution to the refractive index of seawater

The refractive index of seawater (n) is bound together with the n² - law for radiance that corresponds to the geometrical part of the radiance transmittance. However, the notation ‘n’ collectively represents the refractive index of the particulate matter and the pure seawater. In practice, the refractive index of pure seawater [particularly in Eq. (3)] is generally assumed for most oceanic applications [13,17–19]. Since the constant pure water refractive index value only partially accounts for the water-air interface optical process, it is not suitable for applications in the nearshore and inland environments with significant particulate loads [20–24]. Thus, determination of the particulate refractive index is necessary and it is generally deduced relative to pure seawater [23,24]. In this context, the refractive index of seawater is defined as the product of RI of pure seawater n_w(λ) and RI of particulate matter relative to pure seawater r_f, which is expressed as a wavelength-dependent form

3. Theoretical radiance transmittance for natural waters

The theoretically deduced Eq. (7) successfully replaces the conventional pure water geometrical upwelling radiance transmittance with the ‘actual’ upwelling radiance transmittance. Combining Eqs. (7) and (8), we obtain,

4. Results and discussion

From the new theory, it is intuitively understood that the radiance transmittance is not a constant value, but it is strongly dependent on the geometry and multiple interactions of the radiance with the water-air interface. The existing geometrical upwelling radiance transmittance equation $\left(1-{\rho }_{w,a}\right)/n_w^2$ partially accounts for the refractive index of natural water (since it considers the pure water RI), and hence, its application is restricted to open ocean waters. The new radiance transmittance formulation is derived theoretically by incorporating the additional influences due to the refractive index of particles and the multiple interactions of the photons with the interface. Thus, the new formulation of radiance transmittance [Eq. (10)] is applicable for most natural waters.

4.1 Water-air interface and the upward radiances

To determine the water-leaving radiance L_w(air,λ) from the upwelling radiance L_u(0^-,λ), accurate quantification of τ_w,_a is necessary. This requires a better understanding of the behaviour of upward scattered photons emerging from the water. Upward travelling photons have the greater probability of escaping the interface unless they are redirected back to the water medium by the Fresnel reflection on the water side and the downward scattering by the water molecules and particles. The upward radiance is dependent on the incoming solar (downwelling) flux penetrating into the water and is the result of upward scattering by the molecules and the particles present in it. A collection of upward scattered photons within a solid angle is known as the radiance obeys $\left(1-{\rho }_{w,a}\right)/n_w^2$ law and crosses the interface, while the rest of the photons are redirected back to the water medium. The redirected photons are presumably diverted to any direction within the lower hemisphere of the water surface, and sometimes depend on the sea state. It should be pointed out that the directionality of the photon may depend on the sea-state [29], but the $\left(1-{\rho }_{w,a}\right)$ remains unaffected [28]. The redirected photons from the interface cannot freely travel along the direction to which it was focussed, but have interactions with all the particles and water molecules along its path and get modified by the absorption and scattering processes. Few parts of photon energy are lost by absorption and the rest are scattered into other directions according to the shape of the volume scattering function. However, the probability for a (redirected) photon that is scattered upward depends upon two factors – single scattering albedo ‘ω’ and upwelling average cosine ‘µ_u’. The single scattering albedo $\omega (=b/c)$ provides the probability for a photon being scattered rather than absorbed, and has the ability to describe the type of the particle but not the scattered direction. For the scattering direction to be known after single scattering, the average cosine of the light field ‘µ’ provides a better estimate of the photon direction. As our interest lies in the photon that is scattered upward, the upwelling average cosine for the upward light field µ_u gives the probability of photon direction. The upwelling light field originates from upward scattering events at lower depths, which is completely diffuse and can be assumed to be isotropic. Therefore, the average cosine for the upwelling light field is considered as 0.5 [30,31].

Comprehensively, the probability of a scattered photon is signified by ω and µ_uω is the probability of the upward scattered photon. The new terms $(1-{\mu }_u\omega )/r_f{}^2$ and ${\mu }_u\omega n_w^2/(1-{\rho }_{w,a})$ in Eq. (7) are highly responsible for the multiple interactions of photons with the interface, which allow some extra photons to cross the interface in addition to the $\left(1-{\rho }_{w,a}\right)/n_w^2$. $\left(1-{\rho }_{w,a}\right)/n_w^2$ is the pure water geometrical upwelling radiance transmittance, a percentile estimate of the photons crossing the interface based on the geometrical radiometry for the pure water condition, and holds the value of 0.541 (or 54.1%). The particulate refractive index r_f, however, an inclusive part of the geometry, is described relative to the pure water geometrical radiance transmittance τ_w,a. The ratio between the ‘actual’ radiance transmittance τ_w,a and the pure water geometrical radiance transmittance τ_pw,a is equivalent to $\left(1-{\mu }_u\omega \right)/r_f^2+{\mu }_u\omega n_w^2/(1-{\rho }_{w,a})$.

4.2 Examination of theoretical upwelling radiance transmittance with in situ data

We now examine the theoretically derived radiance transmittance τ_w,a [Eq. (7)] with the in situ data. To examine any improvement and difference between the old and the new transmittance formulations, we intended to select the data from various water bodies which are unique and different from each other in their spectral characteristics. Based on this criterion, fourteen data were selected from different stations in clear oceanic waters of southern Bay of Bengal, relatively clear waters of coastal Chennai (slightly increased turbidity as compared to the clear oceanic waters), sediment-dominated waters off Point Calimere, and turbid productive inland waters of Muttukaadu lagoon. These data are optically different from each other that can be recognized from the spectral plots of ω [Fig. 2].

 

Fig. 2 In situ measured data of single scattering albedo $\omega (=b/c )$ for 14 different environments representing the clear oceanic waters (deep blue) to highly turbid scattering waters (red). This plot clearly shows the effects of absorption and scattering with respect to the wavelength. The lowest and highest values of ω measured being near zero for clear open oceanic waters (at a wavelength 750 nm) and 0.97 for mineral rich turbid waters (at a wavelength 715 nm).

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The difference between the τ_w,a and τ_pw,a mainly comes from the three independent factors µ_u, ω, and r_f. Assuming µ_u = 0.5, ω from b/c via in situ measurements, and r_f from the existing model (shown in Appendix B), the new radiance transmittance τ_w,a is calculated. As µ_u is assumed as constant, the variation in τ_w,a is due to the ω and r_f values. The ω is a spectrally dependent parameter and highly sensitive to the water types. Figure 2 gives the trend of ω depicting the spectral variations for 14 different water types. ω is low for clear oceanic waters (around 0.4 in the blue region and nearly zero in the red region), indicating that the scattering probability is relatively high in the blue bands and low in the red bands due to the dominance of water molecules. In contrast, the value of ω increases with the increasing particulate concentration in other water types, as shown by various colors for clear open ocean waters, coastal waters, CDOM and phytoplankton dominant harbour waters, phytoplankton dominant inland waters and mineral sediment dominant coastal waters. Low values of ω indicate the tendency of the particles being inclined towards the absorption and high values towards the scattering. Much variations in the magnitude of ω are noticed at longer wavelengths than shorter wavelengths because of the increased water absorption and particulate scattering. The larger water content in the case of clear oceanic waters causes the water absorption to dominate at longer wavelengths, and the relatively lesser water content in particle-rich coastal and inland waters causes the scattering to dominate the optical processes at these wavelengths. The value of ω goes very high, nearly close to 1, especially in mineral-rich sediment waters because of the high particulate scattering events.

Most importantly, it is observed that the variation in the spectra of ω is small for high scattering waters and large for clear oceanic waters. This reveals that the spectral variations in ω have direct effects on the radiance transmittance τ_w,a. For instance, ω for clear oceanic waters is approximately zero at longer wavelengths, with the particulate refractive index r_f closer to 1, and therefore, µ_uω reaches to zero, resulting the new term $\left(1-{\mu }_u\omega \right)/r_f^2+{\mu }_u\omega n_w^2/(1-{\rho }_{w,a})$ to 1. This shows that typically no photons actually emerge at longer wavelengths in the case of clear oceanic waters, and therefore, τ_w,a is almost equivalent to the pure water geometrical upwelling radiance transmittance, τ_pw,a. However, in the case of high scattering waters with ω = 0.95, µ_u = 0.5 and r_f = 1.1 (for mineral waters [23]), the value of µ_uω is 0.475 resulting the new term to 1.31, and therefore, τ_w,a equals the value of 0.709 which is 1.31 times higher than the τ_pw,a. Thus, by the process of multiple interactions of the photons with the interface, more number of photons are expected to escape through the interface.

In short, the additional term $\left(1-{\mu }_u\omega \right)/r_f^2+{\mu }_u\omega n_w^2/(1-{\rho }_{w,a})$ describes the considerable role of particles on the upwelling radiance transmittance. From this it is clearly evident that the multiple interaction of the photons with the interface always enhances the τ_w,a to be greater than 0.541. As evident in Fig. 3(a), the value of τ_w,a also seems to fall below 0.541 (denoted by the dotted line) at some spectral domains that are inclined toward the absorption. The fall of τ_w,a below 0.541 is due to the refractive index of particles ‘r_f’ which tend to trap the photon within (its medium) itself. According to our investigation, two types of waters can have the probability of τ_w,a falling below the value 0.541: (i) waters with the relatively larger water content characterized by strong absorption of the water molecules particularly at longer wavelengths, and (ii) waters with dominant phytoplankton and/or CDOM which are inclined toward strong absorption particularly in the shorter wavelengths. The higher τ_w,a values are caused by scattering particles as it can be seen at the longer wavelengths [Fig. 3(b)]. τ_w,a of high scattering waters is nearly constant throughout the spectrum, even reaching beyond 0.72. Interestingly, τ_w,a is also much higher in turbid productive waters (a dark brown color spectrum, with high contents of phytoplankton, detritus, mineral sediments, and CDOM) similar to the purely mineral sediment dominated water. The findings reveal that r_f seems to play a major role in determining the magnitude of τ_w,a. While the two mineral dominant stations have the r_f values of 1.139 and 1.151, the turbid productive water station with mixed concentrations of phytoplankton, mineral sediments and detritus is characterized by r_f = 1.07 (see these station spectra located nearby). Since the r_f values of the two mineral dominant water stations (1.139 and 1.151) are greater than the r_f value of the well-mixed turbid productive water station (1.07), the low r_f allows more photons to transmit through the interface and hence the high τ_w,a values for these waters.

 

Fig. 3 (a) Plot of the additional term $\left(1-{\mu }_u\omega \right)/r_f^2+{\mu }_u\omega n_w^2/(1-{\rho }_{w,a})$ representing the ratio of new theoretical upwelling radiance transmittance τ_w,a relative to the pure water geometrical upwelling radiance transmittance τ_w,a (The black dots in ‘a’ represent τ_w,a:τ_pw,a = 1). (b) The new theoretical upwelling radiance transmittance τ_w,a is applied for the natural water types with measured ω values ranging from 0 to 1 (The black dots in ‘b’ represent ${\tau }_{pw,a}=(1-{\rho }_{w,a})/n_w^2\approx 0.541$ line). The legends of these spectra are given in Fig. 2.

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4.3. Conversion of sub-surface r_rs to above-surface R_rs

Remote sensing reflectance (R_rs) is the ratio of water-leaving radiance (L_w) to the incoming downwelling irradiances (E_d(0⁺, λ)) measured above the water surface [Eq. (14)]. Similarly, sub-surface reflectance (r_rs) can be calculated as the ratio of upwelling radiances (L_u(0^-,λ)) (measured at nadir position) to the downwelling irradiances (E_d(0^-,λ)) below the water surface [Eq. (15)].

4.4. Improvement in the ocean color quantities, L_w and R_rs

Improvement of the new theoretical formulation [Eq. (10)] over the constant upwelling radiance transmittance ( = 0.541) can be seen by comparing the 0.541 (dotted) line and other spectra in Fig. 3(a)-(b) and the percentage of improvement in Fig. 4. Based on the assessment of the results in Fig. 3(a)-(b), the deviation of the present formulation τ_w,a with the globally considered constant value 0.541 is minimum for the waters characterized by the weak scattering components (CDOM and phytoplankton dominant) and maximum for the waters characterized by the strong scattering components (mineral-rich sediment dominated waters). Any improvement in the τ_w,a is consequently reflected on the water-leaving radiance (L_w) and remote sensing reflectance (R_rs) spectra with the same percentage of improvement.

 

Fig. 4 Spectral relative percentage of improvement in the ocean color quantities, L_w and R_rs.

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For oceanic waters (blue to pale green waters), improvement in L_w and R_rs spectra is found from 0% to 18% for the wavelength range 400-600nm and goes up to −18% for the wavelength range 600-750 nm, respectively. Here, the (-) minus sign does not denote under performance, but it infers the deviation from the standard value. The harbour waters with high CDOM content and phytoplankton concentration showed up to 22% for the wavelength range 400-750, while the well-mixed turbid productive water with equal parts of phytoplankton and mineral sediments had the range from 20% to 33% for the entire visible wavelengths 400-700nm. The percentage of improvement in high scattering mineral sediment waters varies from 20% to 30% for the entire visible wavelengths. The comparison of the ocean color quantities L_w and R_rs determined using the conventional method and the new method Eqs. (11) and (18) is shown in Figs. 5 and 6. The results show significant improvement in the L_w and R_rs values.

 

Fig. 5 Water-leaving radiance (L_w) determined using the conventional radiance transmittance (grey line) and new theoretical radiance transmittance (dark line) values for four different water types (clear oceanic water – (a), harbour water – (b), turbid coastal water (mineral) – (c), and turbid productive inland water – (d)). Water-leaving radiance (L_w) determined using the conventional radiance transmittance (grey line) and new theoretical radiance transmittance (dark line) values for four different water types (clear oceanic water, harbour water, turbid coastal water (mineral), and turbid productive inland water).

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Fig. 6 Remote sensing reflectances (R_rs) determined using the conventional radiance transmittance (grey line) and new theoretical radiance transmittance (dark line) values for four different water types (clear oceanic water – (a), harbour water – (b), turbid coastal water (mineral) – (c), and turbid productive inland water – (d)).

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4.5. Comparison of the new theoretical formulation with existing models

Recently, Wei et al. [33] examined the long-standing presumption of constant upwelling radiance transmittance based on their in situ measurements of L_w(air, λ) and L_u(0^-,λ) by adopting the skylight-blocked approach. They studied the relative percentage difference of the transmittance $(=L_w(air,\lambda )/L_u(0^{-},\lambda ))$ and the constant 0.54 for the three different water types of blue, turquoise blue and green color. Their τ_w,a for the blue waters showed the minimum value of 0.4 and maximum value of 0.6 and the mean percentage difference approximately 5% for the 350-600 nm range and ± 20% for the 600-700 nm range. The conclusion of their results that yielded −20% in the red bands for blue waters is mainly due to the discrepancies and errors. According to our observations on their data, the −20% difference in the red bands is acceptable as only few photons are expected to cross the interface due to the strong absorption by water molecules. Therefore, such a high difference is possible (not mainly due to errors) and it is consistent with the present study. As expected, the turquoise blue and green waters showed less deviations (within 6%), whereas our results are in agreement with the difference of 0-10% for similar water types.

Lee et al. [34] presented the relation between R_rs and r_rs with the constant coefficients [Eq. (24) and (25)], a semi-analytically modified version of Gordon & Clarke and Gordon et al. [32,35]. The term $(1-{\rho }_{a,w})(1-{\rho }_{w,a})/n_w^2$ of the present study and ζ of Lee et al. [34] are analogous to each other and gives the value of 0.518 with the assumed spectrally constant n_w value. However, the difference arises from the next term $\left(1-{\mu }_u\omega \right)/r_f^2+{\mu }_u\omega n_w^2/(1-{\rho }_{w,a})$ and $1/(1-\text{Γ}{r}_{rs})$ in which the former term describes the contribution of particulate RI and the multiple interactions of the photons with the interface from the water side and the latter term presents some generalized effects of internal reflection from water to air.

According to Eq. (19), the transmission coefficient for a sub-surface remote sensing signal is τ_w,a(1-ρ_a,w). τ_w,a(1-ρ_a,w) for the previous models are approximately closer to 0.54, whereas the present study gives 0 to 0.75 for varying ω and r_f values. Figure 7 shows the approximate range of variations in the transmittance for a sub-surface remote sensing signal of Lee et al. [34] and the present study [Eq. (18)]. The model of Lee et al. [34] provides almost constant transmittance, τ_w,a(1-ρ_a,w)~0.54, as shown in red line, whereas the present study provides a wide range of τ_w,a(1-ρ_a,w) values from 0.40 to 0.75. The values below the red line correspond to the waters inclined toward absorption (blue shaded region) and the values above the red line are inclined toward the scattering (orange shaded region). Comparing both the formulations, the new theoretical formulation successfully takes into account the two key optical processes occurring within the interface for predicting a wide range of transmittance values for natural waters, which enable more accurate remote sensing reflectances for various water color applications.

 

Fig. 7 Comparison of the conversion of sub-surface remote sensing reflectance, r_rs(0^-,λ), to above-surface remote sensing reflectance, R_rs(0⁺,λ), using the present formulation [Eq. (18)] (derived as a function of ω and r_f) and the model of Lee et al. [34] (shown in red line). The lowest line represents the transmission factor for a sub-surface remote sensing signal, τ_w,a(1-ρ_a,w) = 0.40, and the highest line represents the τ_w,a(1-ρ_a,w) = 0.75. The blue and orange shaded regions depict the waters inclined toward absorption and scattering respectively.

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5. Summary and conclusion

The present work replaces the standard upwelling radiance transmittance with the new theoretical upwelling radiance transmittance formulation with potential applications in all natural waters. This new expression serves the purpose of translating the in-water radiance to above surface radiance through the water-air interface, accounting all optical variations from low to high scattering and low to high absorbing waters. The new formulation is exact, in which the geometric conditions and optical phenomenon occurring at the water-air interface including the multiple interactions of photons and the particle optical properties are acknowledged. The results show that the inclusion of new parameters to the geometrical radiance transmittance has led to significant improvement in the accuracy of the determination of the water-leaving radiances and remote sensing reflectances from the below-water radiance measurements (significantly from 0 to 33% depending upon the water types). The percentage improvement is high for the scattering waters and low for the absorbing waters. This study will be important for accurately determining the upwelling radiances that cross the water-air interface in marine and inland environments, where the former approaches are inadequate considering the different types of waters. The new theoretical method will have direct implications for improving the radiometric water color measurements and calibration and validation of space-borne ocean color sensors.

Appendix A n² - law for radiance

The n² - law for radiance is an important law of geometrical radiometry which describes the transfer of light rays from one medium to another medium. Let us assume a ray of light incident on the plane P with the angle θ₁ in the medium n₁ and refracted with the angle θ₂ in the medium n₂ [Fig. 1]. We use the suffixes 1 and 2 to denote the incident and refracted mediums, respectively. According to the Snell’s law, the ratio of the sines of the angles of incidence and refraction of a light ray is equal to the ratio of the refractive indices between them (for example, air-water),

(20)$$n_1\sin {\theta }_1=n_2\sin {\theta }_2.$$

Squaring Eq. (20) on both sides, we get,

(21)$$n_1^2{\sin }^2{\theta }_1=n_2^2{\sin }^2{\theta }_2.$$

Differentiating the Eq. (21),

(22)$$2n_1^2\sin {\theta }_1\cdot \cos {\theta }_1\text{Δ}{\theta }_1=2n_2^2\sin {\theta }_2\cdot \cos {\theta }_2\text{Δ}{\theta }_2.$$

Multiplying $\text{Δ}\varphi$ on both sides, we get,

(23)$$n_1^2\sin {\theta }_1\cdot \cos {\theta }_1\text{Δ}{\theta }_1\text{Δ}\varphi =n_2^2\sin {\theta }_2\cdot \cos {\theta }_2\text{Δ}{\theta }_2\text{Δ}\varphi .$$

The element of solid angle in spherical coordinates is given by, $\text{ΔΩ}=\frac{\text{Δ}A}{r^2}=\frac{(r\sin \theta \text{Δ}\varphi )(r\text{Δ}\theta )}{r^2}=\sin \theta \text{Δ}\theta \text{Δ}\varphi$, therefore,

(24)$${\text{ΔΩ}}_1=\sin {\theta }_1\text{Δ}{\theta }_1\text{Δ}\varphi .$$

(25)$${\text{ΔΩ}}_2=\sin {\theta }_2\text{Δ}{\theta }_2\text{Δ}\varphi .$$

Substituting Eqs. (24) and (25) in Eq. (23), we get,

(26)$$n_1^2\cos {\theta }_1{\text{ΔΩ}}_1=n_2^2\cos {\theta }_2{\text{ΔΩ}}_2.$$

The above equation is known as Straubel’s invariant in geometric optics. According to Eq. (1), the spectral radiance is the measure of radiant power per unit area, per unit solid angle within a wavelength interval. Therefore, it can be rewritten as,

(27)$$L_1=\frac{{\text{ΔΦ}}_1}{\text{Δ}{A}_1{\text{ΔΩ}}_1\text{Δ}\lambda }.$$

(28)$$L_2=\frac{{\text{ΔΦ}}_2}{\text{Δ}{A}_2{\text{ΔΩ}}_2\text{Δ}\lambda }.$$

From the work of Fresnel, we can write the ratio of radiant powers of the respective mediums as equivalent to Fresnel transmittance ‘t’. Therefore, we can write,

(29)$$t=\frac{{\text{ΔΦ}}_2}{{\text{ΔΦ}}_1}.$$

From the immersion optics,

(30)$$\text{Δ}{A}_1=\text{Δ}A\cos {\theta }_1.$$

(31)$$\text{Δ}{A}_2=\text{Δ}A\cos {\theta }_2.$$

Taking ratios of Eqs. (27) and (28),

(32)$$\begin{array}{l}\frac{L_2}{L_1}=\frac{{\text{ΔΦ}}_2 \text{Δ}{A}_1 {\text{ΔΩ}}_1}{{\text{ΔΦ}}_1 \text{Δ}{A}_2 {\text{ΔΩ}}_2}.\\ \frac{L_2}{L_1}=t\frac{\cos {\theta }_1{\text{ΔΩ}}_1}{\cos {\theta }_2{\text{ΔΩ}}_2}.\end{array}$$

Comparing Eqs. (26) and (32),

(33)$$\frac{L_2}{L_1}=t\frac{n_2^2}{n_1^2}.$$

Equation (33) is known as n² law for radiance.

As we deal with the transfer of radiance from water (medium 1) to air (medium 2), then L₁ = L_u, L₂ = L_w and τ = τ_w,a, n₁ = n and n₂ = n_a, where n_a is approximately equal to 1. Then,

(34)$$L_w=\frac{t}{n^2}{L}_u.$$

where t = 1-ρ and t/n² is the radiance transmittance τ. The above form Eq. (34) is represented in Eq. (3) in the manuscript.

Appendix B Calculation of particulate refractive index relative to seawater, r_f

Particulate refractive index r_f is not a direct measured quantity, however, it can be calculated indirectly from the existing models. We have calculated the r_f using Eq. (21) of Sahu and Shanmugam [27]:

(35)$$r_f=1+{\left(\frac{P_1}{{\tilde{b}}_{bp}}\right)}^{\frac{1}{2P_2}}.$$

where, $P_m=a_m{\gamma }^2+b_m\gamma +c_m$; m = 1, 2. a_m, b_m and c_m are the coefficients from Table 2; γ can be obtained from two methods, (i) from the attenuation hyperbolic slope [36,37], or (ii) from the particle size distribution slope (PSD), $\gamma =\xi -3$ [23]. Petzold average-particle phase function can be used as a substitute to the backscattering ratio ${\tilde{b}}_{bp}$ because each phase function has a backscatter fraction of 0.0183 [38].

Table 2. Coefficients for the determination of r_f.

View Table | View all tables in this article

Funding

Department of Science and Technology (DST) through the Networked Project on Big Data Analytics-Hyperspectral Data (OEC1617130DSTXPSHA); Research Management Fund (RMF) from IIT Madras.

Acknowledgments

We sincerely thank the three anonymous reviewers for their helpful and constructive comments.

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