Analysis of the Bidirectional Characteristic of Radiation of Flat and Rough Water–Air Interfaces Based on the Theory of Radiative Transfer

Abstract

The Lambertian property of objects is one of the basic hypotheses in remote sensing research. However, the spectral radiance of natural objects is always anisotropic. On the sea surface, a large amount of sea foam is generated at the water–air interface, induced by wind speed and breaking gravity waves. Additionally, the scattering characteristic at the water–air interface significantly influences the accuracy of ocean color remote sensing and its output. The bidirectionality of the water light field is one of the sources of errors in ocean color inversion. Therefore, the knowledge of the bidirectional reflectance distribution of water surfaces is of great significance in quantitative remote sensing or for the evaluation of measurement errors in surface optical parameters. To clarify the bidirectional reflectance distribution, we used the coupled ocean–atmosphere radiative transfer (COART) model to simulate the bidirectional radiance of water bodies and explored the anisotropy of radiance at the water–air interface. The results indicate that the downward and upward irradiance just below the water surface and the water-leaving radiance changed with the sun-viewing geometry. The downward and upward radiance just below the water surface decreased as the zenith angle of the incident light increased. This effect can be mitigated using a function of the viewing angle. Additionally, the viewing azimuth angle and rough sea surface had no significant effect on the downward and upward radiance. The water-leaving radiance had an obvious bidirectional reflectance characteristic. Additionally, a backward hotspot was found in the simulated results. Then, the transmission coefficient was calculated, and the bidirectional distribution characteristic was found for flat and rough sea surfaces. This study can be used as a reference to correct bidirectional errors and to guide the spectral measurements of water and its error control for rough sea surfaces.

1. Introduction

The Lambertian property of objects is considered one of the basic hypotheses in remote sensing research, and this assumption can satisfy the accuracy needs of the original analyses of remote sensing applications, such as the identification of land use types, change detection, and disaster monitoring [1,2,3,4,5,6]. In studies focusing on classification methods, automatic recognition algorithms, or methods reducing the spatial-scale effect, the non-Lambertian reflectance characteristic of objects is not considered a major factor. For simplicity, many researchers assume the ocean surface to be flat [7,8]. However, the non-Lambertian property of natural objects cannot be ignored when the application of remote sensing technology involves quantitative remote sensing [9]. This is because the wind always generates waves on the surface of the ocean, and the minute differences in the radiance between objects are important in realizing parameter inversion using remote sensing [10,11]. The influence of these factors cannot be ignored, especially for quantitative remote sensing of the ocean. Therefore, an understanding of the real reflectance or emission characteristics of natural oceans is of great significance in quantitative remote-sensing-related research [12,13].

Nicodemus was one of the earliest researchers to propose the application of the bidirectional reflectance distribution function (BRDF) for describing the directional reflectance characteristics of opaque surfaces and provided the base parameters for the BRDF in subsequent studies [14]. However, it was not until the 1980s when research in this field gradually rose to prominence, and the field has made great progress in the past 30 years, with the emergence of various complicated models with their own advantages [15]. Additionally, there are many studies on the bidirectional reflectance characteristics of vegetation and soil on land surfaces, and several remote sensing models considering bidirectional reflectance characteristics have been developed. The geometric–optical model proposed by Li et al. [16,17], the four-scale model proposed by Chen et al. [18], and the discrete anisotropic radiative transfer (DART) model proposed by Gastellu-Etchegorry et al. [19] can all effectively describe the directional reflectance characteristics of targets. The findings of these studies enrich our knowledge of related fields and promote the use of quantitative remote sensing technology [20].

As the main branch of remote sensing technology, ocean color remote sensing plays an important role in research on the ocean carbon cycle, ocean environmental detection, ocean pollution monitoring, etc. [21,22,23,24]. On the sea surface, a large amount of sea foam is generated at the water–air interface, induced by wind speed and breaking gravity waves. Additionally, the scattering characteristic at the water–air interface significantly influences the accuracy of ocean color remote sensing and its output [25]. Therefore, the non-Lambertian characteristics of water bodies are an important topic of research [26,27]. Gordon et al. used a Monte Carlo method to simulate the radiative transfer in natural water bodies and examined the changes in the diffused reflectance with a varying solar zenith angle. The results indicated that the changes were related to the volume-scattering function. At the same time, Gordon noted that the effect of departures from Lambert’s law of diffuse reflection should be considered in some cases [28]. Morel et al. described the spatial distribution characteristics of upward radiance immediately below the water surface by using the Q-factor method for the first time and presented their conclusions on the bidirectional reflectance characteristics of water bodies [29,30]. These two simulated sets of data achieved satisfactory results using the Monte Carlo method, and the measured Q-value method had excellent consistency and validated the feasibility and accuracy of the Monte Carlo approach in simulating the radiative transfer in anisotropic water bodies. Morel et al. simulated the water molecules and particles in the scattering process in oceanic case 1 waters by using a Monte Carlo model and analyzed the diffuse reflectance beneath the surface and its variations with the solar zenith angle. They concluded that molecular scattering is not negligible in most ocean waters [31]. Maritorena [32] analyzed the effects of seabed reflection on diffused reflectance. However, these results were achieved with the assumption of a flat ocean surface. Jin et al. introduced a coupled ocean–atmosphere radiative transfer (COART) model for measuring wind-induced rough sea surfaces and discussed the spatial distribution characteristics of radiation energy immediately above the water surface at different wind speeds [33]. Tang et al. designed a 3D Monte Carlo model to discuss the effects of the solar zenith angle and wind velocity on the spatial distribution characteristics of Q-values [34]. Ling used a Monte Carlo model to simulate the BRDF of water bodies [35]. Chen et al. simulated the directional scattering characteristic of sea surfaces covered in oil pollution [36].

In a previous study, bidirectional reflectance characteristics were applied to the quantitative inversion of land surface parameters as prior knowledge [37,38,39]. In ocean color remote sensing, only radiation energy, which enters the water body and is reflected from the water–air interface and then collected by sensors, can be used for identifying or inverting the features of water bodies. Therefore, it is of considerable significance to study the spatial distribution characteristics of remote sensing reflectance and water-leaving radiance at the water–air interface.

Thus, the main objective of this paper was to analyze the spatial distribution characteristics of remote sensing reflectance and water-leaving radiance at the water–air interface and understand the variation characteristics of water-leaving radiance with the solar zenith angle, taking into account zenith and azimuth angles. To this end, observations or measurements of data with various viewing angles were essential for this study. However, the measurements of real data are restricted by various external factors, such as the characteristics of water bodies, sea winds, rough sea surfaces, shade from boats, sun glint, and sky conditions [40,41,42,43]. These factors will increase the uncertainty in measurement and may lead to errors in datasets and failure in results. By contrast, radiative transfer models can simulate reliable and repeatable datasets; radiative transfer models are therefore efficient tools to study the bidirectional radiation characteristics of water bodies.

To this aim, we used a COART model to simulate the upward radiance immediately below the water surface and the water-leaving radiance at different observation geometries and wind velocities. Additionally, the effects of a rough sea surface on the spatial distribution characteristics of the radiance were evaluated. This study provides a better understanding of the directional reflectance characteristics of water–air interfaces, especially for rough sea surfaces, and elucidates the effect of rough sea surfaces on ocean color remote sensing inversion. In addition, this study can be used as a reference for the spectral measurement of water and its error control for rough sea surfaces.

2. Methodology

2.1. Method

The water-leaving radiance immediately above the water surface L_w(0⁺), remote sensing reflectance R_rs, and diffused reflectance immediately below the water surface R are indicators of the optical properties of ocean color and are detected using ocean color satellite sensors. These parameters that are inversed using atmospheric correction can then be used to estimate the optical properties of water bodies [44]. Therefore, it is of great significance to evaluate the effects of the bidirectional reflectance characteristics of the radiance at the water–air surface.

The theoretical radiative transfer formulations in the coupled atmosphere–ocean system using the discrete ordinate method are described as Equation (1) [33]:

$$\mu \frac{dI(\tau ,\mu ,\phi )}{d\tau }=I(\tau ,\mu ,\phi )-S(\tau ,\mu ,\phi )$$

(1)

where $I(\tau ,\mu ,\phi )$ is the radiance in direction $(\mu ,\phi )$ at optical depth $\tau$; $S(\tau ,\mu ,\phi )$ is defined as the source function; $\mu$ is the cosine of the zenith angle (θ) such that $\mu \in [-1,1]$; $\phi$ is the azimuth angle; and $S(\tau ,\mu ,\phi )$ is the actual internal source Q plus the scattering in direction µ,φ from all other directions [45]. And $S(\tau ,\mu ,\phi )$ is described as Equation (2):

$$S(\tau ,\mu ,\phi )=Q(\tau ,\mu ,\phi )-\frac{\omega (\tau )}{4\pi }{\displaystyle \underset{0}{\overset{2\pi }{\mathrm{\int }}}d{\phi }^{\prime }}{\displaystyle \underset{-1}{\overset{1}{\mathsf{\int }}}d{\mu }^{\prime }p(\tau ,\mu ,\phi ,{\mu }^{\prime },{\phi }^{\prime })I(\tau ,{\mu }^{\prime },{\phi }^{\prime })}$$

(2)

where $\omega (\tau )$ is the single-scattering albedo; $p(\tau ,\mu ,\phi ,{\mu }^{\prime },{\phi }^{\prime })$ is the phase function for the incident direction $({\mu }^{\prime },{\phi }^{\prime })$ and scattering direction $(\mu ,\phi )$; and $Q(\tau ,\mu ,\phi )$ is the source term which will be different for flat and rough ocean surfaces.

$I(\tau ,\mu ,\phi )$, which is the radiance in direction $(\mu ,\phi )$, can be expanded into a Fourier cosine series of 2N. Additionally, the phase function $p(\tau ,\mu ,\phi ,{\mu }^{\prime },{\phi }^{\prime })$ can also be expanded into a Legendre polynomial series of 2N. Further, the equations for each azimuth radiance component in the atmosphere and ocean can be derived, which are described as follows:

$$\left\{\begin{array}{c}{\mu }_i^a\frac{dI(\tau ,{\mu }_i^a)}{d\tau }=I(\tau ,{\mu }_i^a)-{\displaystyle \sum _{j=-N_1}^{N_1}{\omega }_j^a}D(\tau ,{\mu }_i^a,{\mu }_j^a)I(\tau ,{\mu }_j^a)+X_0(\tau ,{\mu }_i^a)\exp (\frac{-\tau }{{\mu }_0}),(\tau \le {\tau }_a j\ne 0), \mathrm{atmosphere}\\ {\mu }_k^o\frac{dI(\tau ,{\mu }_k^o)}{d\tau }=I(\tau ,{\mu }_k^o)-{\displaystyle \sum _{p=-N_2}^{N_2}{\omega }_p^o}D(\tau ,{\mu }_k^o,{\mu }_p^o)I(\tau ,{\mu }_p^o),(\tau >{\tau }_a p\ne 0), \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \text{ocean}\hfill \end{array}\right.$$

(3)

In Equation (3), $i=\pm 1,\pm 2,\dots ,\pm N_1$ and $k=\pm 1,\pm 2,\dots ,\pm N_2$, where 2N₁ and 2N₂ are the numbers of quadrature points applied in the atmosphere and ocean; ${\tau }_a$ is the total optical depth of the atmosphere; $D(\tau ,{\mu }_i^{},{\mu }_j^{})$ and $X_0(\tau ,{\mu }_i^{})$ are the Fourier components of the beam source (details can be found in the publication of Jin and Stamnes (1994) [46]); ${\mu }_i^a$ and ${\mu }_k^o$ are quadrature points for the atmosphere and ocean; and ${\omega }_j^a$ and ${\omega }_p^o$ are the weights of the atmosphere and ocean, respectively.

We denote the optical depth immediately above and below the ocean surface as ${\tau }^{a-}$ and ${\tau }^{a+}$, respectively. The radiance at the air–water interface can be expressed as follows:

$$I({\tau }^{a-},{\mu }_i^a)={\displaystyle \sum _{j=1}^{N_1}I({\tau }^{a-},-{\mu }_j^a)R}({\mu }_i^a,{\mu }_j^a,\frac{n_w}{n_a})+{\displaystyle \sum _{j=1}^{N_2}I({\tau }^{a+},{\mu }_j^o)T}({\mu }_i^a,{\mu }_j^o,\frac{n_a}{n_w})+\frac{1}{\pi }{\mu }_0{F}_0^{}\exp (-\frac{{\tau }^a}{{\mu }_0})R({\mu }_i^a,{\mu }_0^{},\frac{n_w}{n_a})$$

(4)

In Equation (4), $i=1,2,\dots ,N_1$.

$$I({\tau }^{a+},-{\mu }_i^0)={\displaystyle \sum _{j=1}^{N_1}I({\tau }^{a-},-{\mu }_j^a)T}({\mu }_i^o,{\mu }_j^o,\frac{n_w}{n_a})+{\displaystyle \sum _{j=1}^{N_2}I({\tau }^{a+},{\mu }_j^o)R}({\mu }_i^o,{\mu }_j^o,\frac{n_a}{n_w})+\frac{1}{\pi }{\mu }_0{F}_0^{}\exp (-\frac{{\tau }^a}{{\mu }_0})T({\mu }_i^o,{\mu }_0^{},\frac{n_w}{n_a})$$

(5)

In Equation (5), $i=1,2,\dots ,N_2$; $n_w$ and $n_a$ represent the refractive indices of the water and air, respectively; and $R({\mu }_i^{},{\mu }_j^{},n)$ and $T({\mu }_i^{},{\mu }_j^{},n)$ are the reflection and transmission functions for a rough surface, respectively. Light incident on a flat water surface will be directly reflected or refracted. When the light is incident on a rough surface, more photons will be scattered. A surface slope distribution function $p({\mu }^{\prime },{\phi }^{\prime }\to \mu ,\phi )$ is used to describe the effects of the surface waves (details can be found in a previous study by Jin et al. (2006) [33]).

For a flat sea surface, the relationship between the water-leaving radiance L_w(0⁺) and the upward radiance just below the water surface L_u(0⁻) can be expressed as follows [47]:

$$L_w(0^{+},\theta ,\phi )=\frac{[1-\rho (\theta ,{\theta }^{\prime })]}{n^2}\times {\displaystyle \int L_u(0^{-},{\theta }^{\prime },\phi )}$$

(6)

For convenience, the variables of viewing angles were added to Equation (6), where θ and φ represent the observation solar zenith angle and azimuth angle, respectively. The parameter θ′ represents the transmission angle, and θ′ = arcsin(sinθ/n) based on Snell’s law, where n represents the refractive index of the water body. ρ(θ′,θ) represents the Fresnel reflection coefficient of the upward radiance. Additionally, the azimuth angle of the sun used in this paper was defined as 180°.

According to radiative transfer theory, Equation (6) can be simplified as follows [48]:

$$L_w(0^{+},\theta ,\phi )=t({\theta }^{\prime },\theta ,n)\times {\displaystyle \int L_u(0^{-},{\theta }^{\prime },\phi )}$$

(7)

where t(θ′,θ,n) represents the transmission coefficient, which is a function associated with the observation angle.

In the real world, an absolutely flat sea surface does not exist. The relationship between the water-leaving radiance L_w(0⁺) and the upward radiance immediately below the water surface L_u(0⁻) on a rough sea surface can be expressed as follows [33]:

$$L_w(0^{+},\theta ,\phi )=L_u(0^{-},{\theta }^{\prime },\phi )\times t({\theta }^{\prime },\theta ,n)\times p(\theta ,\phi \to {\theta }^{\prime },\phi ,\sigma )\times s(\theta ,{\theta }^{\prime },\sigma )$$

(8)

where t(θ′, θ, n) represents the Fresnel transmission coefficient, and p(θ,φ→θ′,φ,σ) represents the probability density function of the small wavelet surface direction. The small wavelet surface is defined as a tiny fragment which is a set of independent multidirectional fragments fitting the rough sea surface. Additionally, the direction of the small wavelet surface satisfies specific statistical laws. Cox and Munk proposed a Gaussian function to explain the probability density function of the small wavelet surface and described the correlation between the function variance σ² and wind velocity U (m·s⁻¹) [49]. The variable S(θ,θ′,σ) represents the shading coefficient, which is used to describe the shading effect of other wave planes on the reflected light and the multiple scattering characteristics between wave planes.

By consolidating Equations (7) and (8), the expressions for the water-leaving radiance L_w(0⁺) and the upward radiance just below the water surface L_u(0⁻) can be merged into the following common equation(Equation (9)):

$$L_w(0^{+},\theta ,\phi )=L_u(0^{-},{\theta }^{\prime },\phi )\times T(\theta ,\phi \to {\theta }^{\prime },\phi ,n,\sigma )$$

(9)

where $T(\theta ,\phi \to {\theta }^{\prime },\phi ,n,\sigma )$ is defined as the transmission coefficient of the water body, which is a function of the solar zenith angle, the viewing zenith angle, and the azimuth angle, satisfying the following equation (Equation (10)):

$$\left\{\begin{array}{c}T(\theta ,\phi \to {\theta }^{\prime },\phi ,n,\sigma )=t({\theta }^{\prime },\theta ,n)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{flat} \mathrm{surface}\\ T(\theta ,\phi \to {\theta }^{\prime },\phi ,n,\sigma )=t({\theta }^{\prime },\theta ,n)\times p(\theta ,\phi \to {\theta }^{\prime },\phi ,\sigma )\times s(\theta ,{\theta }^{\prime },\sigma )\begin{array}{c}\end{array}\mathrm{rough} \mathrm{surface}\end{array}\right.$$

(10)

When the numerical solution for T(θ,φ→θ′,φ,n) can be derived from a given water-leaving radiance L_w(0⁺,θ,φ) and the corresponding upward radiance just below the water surface L_u(0⁻,θ′,φ), the spatial distribution characteristics of the transmission coefficient and the effects of a rough sea surface on the transmission coefficient can be studied.

2.2. Radiative Transfer Model and Parameter Setting

The coupled ocean–atmosphere radiative transfer (COART) model [50] was developed based on the discrete ordinate radiative transfer (DISORT) model [51]. The advantage of the COART model is that the refraction characteristics of water–air interfaces and the reflectance on wind-induced rough sea surfaces are considered in this model. Additionally, a high accuracy of the simulation can be achieved, compared with other models [52]. It can be used to calculate the upward and downward radiation energy at different atmospheric heights, degrees of sea surface roughness, and ocean depths. Therefore, the COART model was considered a good choice for this study.

The input parameters of the COART model mainly include three types of parameters, namely, the solar zenith angles of the incident light, atmosphere parameters, and oceanic parameters. In this research, the solar zenith angle of the incident light was assumed to be in the range of [0, 90°], changing at increments of 10°. This range was adopted to discuss the effects of the solar zenith angle changes in the incident light on the radiation energy of the sea surface. Several standard aerosol models are listed for selection in the COART model. In this study, a standard atmospheric model was adopted. The oceanic parameters were defined as

Table 1. Oceanic input parameters of COART model.

Parameter	Range	Increment
Wind velocity (m/s)	[0, 12]	2
Water depth (m)	100	-
Bottom albedo	0.1	-
Chlorophyll concentration (mg/m3)	0.5	-
Bottom albedo characteristics	Defaulted	-
Particle scattering function	Defaulted	-
DOM absorption	Defaulted	-

.

For the study of directional reflectance at the water–air interface, many types of measurement are essential. However, field measurements are limited by wind speed, waves, sky conditions, shade from boats, etc. [53]. These factors will lead to immeasurable uncertainty for future analysis. In comparison, simulated data using radiative transfer models can compensate for the shortcomings of measured data. The COART model can be used to simulate the upward and downward radiance on flat and rough sea surfaces, the upward and downward radiance immediately below the water surface, and the water-leaving radiance at different viewing angles. The simulated data were used to determine the change characteristics of the water-leaving radiance with different observation geometries, and the correlation between the wind-induced sea surface roughness and water-leaving radiance was assessed. In addition, the upward radiance just below the water surface and the water-leaving radiance were used to calculate the transmission coefficient of water bodies with a rough sea surface, and to discuss the effects of the viewing angle and wind velocity on transmission coefficients. To simplify the database, a radiance level of 550 nm was only used in this study. The flowchart is shown in Figure 1.

3. Results

3.1. Effects of the Zenith Angle of Incident Light on the Downward and Upward Irradiance just below the Water Surface

The downward irradiance just below the water surface E_d(0⁻) determines the energy entering the water body, and the upward irradiance E_u(0⁻) determines how much energy passes through the water–air interface and is detected by sensors. Therefore, the changes in the downward and upward irradiance just below the water surface with the zenith angle of the incident light were analyzed.

The COART model was used to simulate the downward and upward irradiance just below the water surface at 550 nm with various zenith angles of incident light on a rough sea surface. The scatter plots are shown in Figure 2. The solid and dotted lines represent the downward and upward irradiance, respectively, and the color of the line represents the wind-induced water surface roughness (Figure 2). The downward radiance just below the water surface E_d(0⁻) decreased with an increase in the solar zenith angle. The decreasing trend revealed a strong linear correlation with the solar zenith angle of the incident light, with a coefficient of determination of 0.9493. Figure 2 also indicates that the wind-induced water surface roughness had no influence on the downward radiance immediately below the water surface E_d(0⁻). The upward irradiance just below the water surface at 550nm E_u(0⁻) had similar results. A linear relationship was observed between E_u(0⁻) and the solar zenith angle, as shown in Figure 2, with a coefficient of determination of 0.8636. Additionally, it was found that the changes in the zenith angles would lead to changes in the radiance. This result was similar to those observed in a study by Aas, who revealed that the relative errors of the measured reflectance range from −10% to +1% for changes in the incident light [54]. Additionally, the wind-induced surface roughness had no effect on E_u(0⁻).

The trend of the upward irradiance just above the water surface at 550 nm E_u(0⁺) with various zenith angles of the incident light on a rough sea surface is shown in Figure 3. The change in the upward irradiance just above the water surface was not significant when the solar zenith angle was less than 60°, and E_u(0⁺) was significantly attenuated when the solar zenith angle was larger than 60°. However, the wind-induced surface roughness had no significant effect on E_u(0⁺).

3.2. Spatial Distribution Characteristics of Water-Leaving Radiance

The water-leaving radiance is a variable affected by the observation geometry. To clarify the relationship between the water-leaving radiance and the viewing zenith and azimuth angles, we changed the zenith angle of the incident light from 0 to 80° with increments of 10°, and then the water-leaving radiance of flat and rough sea surfaces at 550 nm was simulated using the COART model.

Figure 4 shows how the water-leaving radiance changes with the viewing zenith angle for the zenith angles of the incident light ranging from 0° to 80°. The water-leaving radiance decreased with an increase in the viewing zenith angle when the zenith angle of the incident light was at 0°. However, at the same viewing zenith angle, the water-leaving radiance did not change with a change in the viewing azimuth angle. Therefore, at the zenith angle of the incident light at 0°, the water-leaving radiance exhibited isotropic characteristics at various viewing azimuth angles. When the solar zenith angle of the incident light was at 20°, the spatial distribution characteristics of the water-leaving radiance revealed bidirectional reflectance characteristics at the water–air interface. The isotropic characteristics of the water-leaving radiance became increasingly significant with the increase in the zenith angle of the incident light. Jin et al. also concluded that the radiance field in the upper layers of the ocean is highly anisotropic and varies with the angle of incidence [33].

Then, we simulated the water-leaving radiance at different wind speed velocities from the zenith angle of incident light from 0 to 80° with increments of 10° for roughness sea surface. The results showed that there were not significant changes among those results at the same observation geometries. To reduce redundancy, only a few results are shown. Figure 5 demonstrates the spatial distribution characteristics of the water-leaving radiance at different wind velocities. Specifically, the figure shows the spatial distribution characteristics of the water-leaving radiance with the solar zenith angles of the incident light set at 0°, 30°, and 60° and wind velocities of 0 m·s⁻¹, 6 m·s⁻¹, and 12 m·s⁻¹. The results indicate that the spatial distribution characteristics of the water-leaving radiance were not affected by the wind-induced sea surface roughness.

Then, we analyzed the changes in the water-leaving radiance at various viewing azimuth angles at a constant solar zenith angle and viewing zenith angle. We performed the analysis thus to ensure good sunlight, because most field measurements are usually executed from 10 am to 14 pm. The solar zenith angle also changed from −30° to 30° during the measurement. Therefore, the solar zenith angle was fixed as 30°. An anisotropic characteristic of the water-leaving radiance was markedly observed at a zenith angle of incident light of 30° (Figure 6). The water-leaving radiance approximating an azimuth angle of 0° was higher than the other directional water-leaving radiance values at the same viewing zenith angle. Additionally, this trend of change in the water-leaving radiance was still noticeable when the viewing zenith angle changed. The only difference was the peak value at a viewing zenith angle of 180.

We calculated the water-leaving radiance at a solar zenith angle of incident light of 30° and nine viewing azimuth angles between 0° and 360° and determined the correlations between the water-leaving radiance and the viewing zenith angle (Figure 7). The results show that the water-leaving radiance increased with an increase in the viewing zenith angle. However, the water-leaving radiance began to rapidly decrease as the viewing zenith angle increased beyond 60°. Our results also indicate that these trends were not affected by the changes in the viewing azimuth angle.

The mean values of the water-leaving radiance data at various azimuth angles were calculated, and a scatter plot of the relationship between the water-leaving radiance and viewing zenith angles was drawn, as shown in Figure 8. As shown in the figure, the water-leaving radiance first very slowly decreased and then increased with the increase in the viewing zenith. When its maximum was reached in the mirror reflection direction, the radiance decreased again. The equation for calculating the water-leaving radiance L_w as a function of the viewing zenith angle θ_r can be expressed as follows (Equation (11)):

L_w(θ_r) = −3.0554 × θ_r³ + 5.317 × θ_r² − 1.9393 × θ_r + 3.8369 R² = 0.9286

(11)

The viewing zenith angle changed in radian units to reduce the difference in magnitude between the water-leaving radiance and the viewing zenith angle. The fitting coefficient of determination was 0.9286, and the fitting root-mean-square error (RMSE) was 0.078566. Additionally, a similar result can be found for other solar zenith angles of the incident light. However, this formula can be used to optimize the water-leaving radiance at a solar zenith angle of incident light of 30°. If we want to optimize the water-leaving radiance at other zenith angles of the incident light, a new fitting equation is needed.

3.3. Transmission Characteristics of Radiance at Water–Air Interface

We defined a transmission coefficient T in Section 2.1, which represents the transmission characteristics of the water body when the light passes through the water–air interface. The COART model was used to simulate the water-leaving radiance and the upward radiance just below the water surface at 550 nm with a chlorophyll content of 0.5 mg·m⁻³ at different incident light angles. Additionally, this dataset was used to calculate the transmission coefficient T, and the correlation between the viewing angle and the transmission coefficient T was determined.

The simulated water-leaving radiance at various viewing zenith angles and the upward radiance immediately below the water surface were used to calculate the transmission coefficient for flat and rough sea surfaces using Equation (9), and the results are shown in Figure 9.

On a flat sea surface, the transmission characteristics of the radiance at the water–air interface did not change with the zenith angle of the incident light (see Group a in Figure 9). At the same incident light angle, the transmission coefficient was not significantly affected by surface roughness (see Figure 9(a1–c1)). At a fixed viewing zenith angle, the effects of wind velocity on the transmission coefficient were analyzed. Additionally, we compared the transmission coefficient at various wind speeds with the transmission coefficient at the flat sea surface, and the changing rate was calculated. The results showed that wind velocity increased from 0 m·s⁻¹ to 12 m·s⁻¹, and the changes in the transmission coefficient were less than 1%, which was compared using the transmission coefficient with the results for flat and rough surfaces (results in Figure 9 b,c compared with those in Figure 9a). Therefore, we concluded that the wind-induced sea surface roughness had no significant impact on the transmission characteristics of the radiance at the water–air interface.

The changes in the transmission coefficient with the viewing zenith angle on a flat surface are shown in Figure 10. The results show that the transmission coefficient T did not change with the change in the viewing azimuth angle. Statistical analysis revealed that the change in the transmission coefficient with the change in the azimuth angle was less than 1%. Figure 10 clearly indicates that the transmission coefficient T was significantly reduced with an increase in the viewing zenith angle.

A scatter plot was drawn between the mean transmission coefficients T under different wind velocities, viewing azimuth angles, and viewing zenith angles (Figure 11). The results show that the various solar zenith angles had no effect on the transmission coefficient. As shown in the plot, the decreasing characteristics of the transmission coefficient appear as a parabolic curve with the increase in the viewing zenith angle θ_v. After obtaining a tangent function of the viewing zenith angle θ_v, the decreasing trend of the transmission coefficient T can be rewritten as a linear function of the viewing zenith angle as follows:

y = −0.0058 × tan²θ_v + 0.547

(12)

In this equation (Equation (12)), y is the transmission coefficient. The fitting coefficient of determination was 0.9871, and the fitting RMSE was 0.0065.

4. Discussion

The contribution of energy to various factors in atmosphere–ocean systems needs to be specified through energy simulation and the determination of factors such as angular and spectral characteristics and the polarization of the light field [55]. At the same time, the reflection characteristics of the target are also very important for understanding the imaging process of remote sensing. In a previous study, the Lambertian property of objects was considered one of the basic hypotheses in remote sensing research. However, the spectral radiance of objects in a natural environment, including bodies of water, is generally not isotropic [56,57,58]. In addition, for simplicity, many studies assume the ocean surface to be flat [7,8]. The assumption of Lambertian properties would, therefore, lead to a significant underestimation of the top-of-atmosphere reflectance [59], which will significantly influence the accuracy of ocean color remote sensing and its output [60]. For an accurate estimation of the properties of a water body, the effects of the anisotropic properties of the sea on the water-leaving radiance should be considered. Additionally, the knowledge of the bidirectional reflectance distribution function of the water body is important for quantitative analysis in ocean color remote sensing research [61]. Radiative transfer models in coupled atmosphere–ocean systems are ideal solutions to better address these issues.

The COART was used to describe the BRDF of the radiance under different light conditions, flat and rough sea surfaces. The results show that the anisotropic property depended on the geometry between the sun and the viewing sensor as well as the optical properties and composition of the water. Our results were similar to those obtained by Zhai et al. and He et al. [62,63]. The downward and upward irradiance just below the water surface at 550 nm decreased with an increase in the solar zenith angle. The discrepancy in the upward irradiance at the ocean surface became larger with the increase in the solar zenith angle [46]. This result is similar to that found in Wang’s research [64]. The reason is that the optical path increased with the increase in the solar zenith angle, and the energy of the light was absorbed more by the water body. The trend of change in the upward irradiance just above the water was different from the irradiance just below the water. It showed no significant change when the solar zenith angle was less than 60° and rapidly decreased when the solar zenith angle was larger than 60°. The main reason is that the upward irradiance was unable to pass through the water–air interface due to the excessive refraction angle when the solar zenith angle increased [34]. Additionally, only a small part of the multiple scattering energy transferred to the air, so the upward irradiance just above the water rapidly decreased. To summarize, the effect of the sun-viewing geometry on irradiance should be considered. Ma et al. investigated the spectral reflectance and the BRDF of the sea foam layer in the visible and near-infrared spectrum and concluded that the effects of subsurface scattering and wind speed on the spectral reflectance and BRDF can be neglected in most cases [25]. Zhai et al. found similar results [62]. They compared the scalar radiance errors caused by the wind speed by using an exact vector (polarized) radiative transfer (VRT) model in coupled atmosphere–ocean systems and found that the wind speed changed the scalar radiance errors; however, these effects were not very significant [65]. Similar results were found in this study. The simulated results indicate that the irradiance was not affected by the wind-induced surface roughness. Interestingly, these results also suggest that wind-induced surface roughness can be ignored when irradiance is obtained.

The water-leaving radiance is an important variable for quantitative remote sensing. We analyzed the bidirectional characteristics at the water–air interface and summarized the trend of change with the sun-viewing geometry. The results indicate that the water-leaving radiance was markedly anisotropic. Meanwhile, a backward hotspot was found in the backscattered region. This results are consistent with those of Smith [66] and Morel [30]. It should be noted that the hotspot mainly came from the anisotropic reflectance, which is very common in anisotropic reflections on the land surface. However, the energy in the hotspot direction is meaningless for ocean color remote sensing, and we should avoid including this part of energy in the sensor’s measurement when applied in the field [67]. Additionally, the effect of sun glint is considered when ocean color remote sensors are designed [68].

We found that the relationship between the water-leaving radiance at different viewing azimuth angles and viewing zenith angles was similar, but the value of the water-leaving radiance was different. The main reason is that the diffusion of the water caused an uneven distribution of energy. This uncertainty caused by the BRDF can be calculated using the relationship between the water-leaving radiance and the viewing azimuth angle established in this study. Wang et al. found that the water-leaving radiance backscattered out of water is reduced with an increase in the solar zenith angle, and this is mainly due to the decrease in the downward irradiance of the incident light into the water [64]. The transmission coefficient was a variable of the viewing zenith angle and independent of the viewing azimuth angle. In addition, the water-leaving radiance and the transmission coefficient were not affected by the wind-induced surface roughness [33]. This does not mean that the radiance received by the remote sensor was not affected by the rough sea surface, because wind can change the reflected solar radiances at the surface, and Rayleigh and aerosol radiances are not enough to compensate for the attenuation [33]. This result highlights the significance of observation geometries when measuring radiation data from water surfaces, as well as in ensuring the acquisition of less error-filled data.

It can be noted that some uncertainty may exist in the current research. The upwelling radiance is affected by salinity, temperature, phytoplankton, sediment, yellow substance, and polarization on top of the atmosphere. The bidirectional reflectance also changes with the increase in the chlorophyll concentration [69]. In coastal regions, inorganically suspended material dominates, and the radiance just above the ocean surface will be more sensitive to changes in yellow substance. The distribution of the particles will change the scattering order and upwelling radiance. Additionally, our results were based on model simulation, and therefore new field measurements are necessary to verify these theoretical results. Thus, future research should take into account the abovementioned factors.

5. Conclusions

Due to the non-isotropic characteristic of the volume-scattering function and illumination conditions above the water surface, water-leaving radiance is not generally isotropic. Additionally, the waves of the water surface also have a significant influence over the anisotropy of the water-leaving radiance field and cannot be ignored for quantitative ocean remote sensing. To investigate this issue, we used the COART model to simulate the upward radiance just below the water surface and the water-leaving radiance at different incident angles and different degrees of sea surface roughness and explored the bidirectional distribution characteristics of the water-leaving radiance and the spatial distribution characteristics of the transmission coefficient at the water–air interface. The results were as follows:

• The downward and upward radiance values just below the water surface were affected by the zenith angle of the incident light, and this error could be corrected by using a related solar zenith angle function.
• The spatial distribution characteristics of the water-leaving radiance were anisotropic with the change in the zenith angle of the incident light. This error could be corrected by using a unary cubic equation of the viewing zenith angle.
• The zenith angle of the incident light, the viewing azimuth angle, and the roughness of the water surface had less effect on the transmission coefficient of the water–air interface. However, it was significantly reduced with changes in the viewing zenith angle.

In this study, we assessed the bidirectional reflectance distribution of different water surfaces. The findings of this study will serve as a reference for BRDF optimization in ocean color remote sensing and evaluations of measurement errors in surface optical parameters under different weather conditions.