Dependence of the Bidirectional Reflectance Distribution Function Factor ƒ′ on the Particulate Backscattering Ratio in an Inland Lake

Abstract

The bidirectional reflectance distribution function (BRDF) factor ƒ′ provides a bridge between the inherent and apparent optical properties (IOPs and AOPs) of inland waters. The previous BRDF studies focused on ocean waters, while few studies examine inland waters. It is meaningful to improve the theory of remote sensing of water surface and the accuracy of image derivation in inland waters. In this study, radiative transfer simulation was applied to calculate the ƒ′ values using appropriate IOPs based on in situ combined with realistic boundary conditions (N = 11,232). This study shows that ƒ′ factor varied over the range of 0.33–16.64 in Lake Nansihu, a finite depth water, higher than the range observed for the ocean (0.3–0.6). Our results demonstrate that the factor ƒ′ depends on not only solar zenith angle (${\theta }_s$) but also the average number of collisions ($\stackrel{-}{n}$) and particulate backscattering ratio (${\tilde{b}}_{bp}$). The ƒ′ factor shows a continuous geometric increase as the solar zenith angle increases at 400–650 nm but is relatively insensitive to solar angle in the 650–750 nm range in which ƒ′ increases as ${\tilde{b}}_{bp}$ and $\stackrel{-}{n}$ decreases. To account for these findings, two empirical models for ƒ′ factor as a function of ${\theta }_s$, $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ are proposed in various spectral wavelengths for Lake Nansihu waters. Our results are crucial for obtaining Hyperspectral normalized reflectance or normalized water-leaving radiance and improving the accuracy of satellite products.

1. Introduction

The ability to model the irradiance distribution in inland waters is a fundamental step in the development of advanced remote sensing techniques that can accurately determine the optically significant components. The irradiance reflectance just below the surface ($R(0^{-})=E_u/E_d$, where $E_u$ and $E_d$ are the upward and downward irradiances just beneath the water surface) contains necessary information about the constituents of the water [1,2], as apparent optical properties (AOPs) are largely determined by inherent optical properties (IOPs). Assessing the relationship between $R(0^{-})$ and IOPs is valuable, as it enables estimation of IOPs from irradiance reflectance, which is more easily measured [2,3], for use in the theory of water color remote sensing. The most widely used remote sensing optical algorithm is expressed as [1,4,5].

$$R(0^{-})=f\frac{b_b}{a}=f^{\prime }\frac{b_b}{a+b_b}$$

(1)

where $a$ and $b_b$ are the total absorption and backscattering coefficients, respectively. $f$ is a distribution factor that depends on the solar zenith angle (${\theta }_s$), wavelength ($\mathsf{\lambda }$), volume scattering functions (VSFs, $\mathsf{\beta }$), surface roughness (via the wind speed, W), and aerosol optical thickness (${\tau }_a$) [2,3,4,6,7,8,9]. The coefficient $f^{\prime }$ should be used if $b_b$ is not small with respect to $a$, and the relationship between $f$ and $f^{\prime }$ is expressed as $f^{\prime }=f(1+a/b_b)$, where $f$ and $f^{\prime }$ are approximately 0.3–0.6 for all natural waters [10]. Variability in the $f$ or $f^{\prime }$ factor and its influence on remote sensing of ocean color have been the focus of several theoretical studies [2,3,6,7,11,12,13]. Kirk [2] proposed an approximation that is a function of the cosine of the in-water sun angle and obtained values in the range of 0.3–0.6; Morel et al. [6] established a lookup table for the $f$ factor, dependent on ($1-\cos {\theta }_s$) and the chlorophyll concentration in open ocean waters. This method was adopted by the National Aeronautics and Space Administration and has become a standard method. However, it works well only with ocean waters and with a maximum chlorophyll concentration of 10 mg/m³, obviously not with inland waters. Previous studies of bidirectional reflectance $f\prime$ factor were restricted to oceanic and coastal waters [6,7,8], but the variation of $f\prime$ factor in inland Lakes remains unknown. The complex behavior of photons in inland waters resulting from the combined effects of the absorption and scattering coefficients, VSFs, and optically shallow bottoms, remains unclear. Further studies are needed for inland waters, which are impacted by human activities and have extremely complex apparent optical properties.

The current $f^{\prime }$ factor models are built using one or two VSFs since the VSFs measurement over broad angular ranges has been uncommon [14,15,16]. Particulate scattering is an important process determining both light penetrations through the water and light leaving the water. Further elucidation of the impact of the particulate backscattering ratio on $f$ or $f^{\prime }$ remains an important challenge. Therefore, investigation of the relationship between the particulate backscattering ratio (${\tilde{b}}_{bp}\equiv b_{bp}/b_p$, where $b_{bp}$ and $b_p$ are particulate scattering and backscattering, respectively) and $f$ or $f^{\prime }$ is very meaningful. As a result, the previous studies had a few focus on the effect of the VSFs on the bidirectional reflectance factor. He et al. [7] developed a bidirectional subsurface remote sensing reflectance model accounting for paticle backscattering shapes for an optically deep ocean, and it was requried to measure VSFs as a input, which was hard to apply the inland lakes because of the bottom reflectance and VSFs measurement. The particulate backscattering ratio, ${\tilde{b}}_{bp}$, supports the derivation of an approximate scattering phase function. As a result, ${\tilde{b}}_{bp}$ has been used to generate a particulate phase function based on Mie theory and the Fournier–Forand phase function [17,18]. This ratio can be assumed to be spectrally constant [19,20]. Surface ${\tilde{b}}_{bp}$ values range between 0.0004 and 0.06 [19,21,22,23,24,25,26,27,28], and particles in lakes are often divided into chlorophyll-bearing particles and mineral particles. According to the information available about the particulate composition and previous studies, 16 values of the chlorophyll-bearing particle ratio, ${\tilde{b}}_{bp}^c$, were selected from 0.005 to 0.018 at an interval of 0.001, as well as 0.0024 and 0.0183 [29]. For the mineral particle ratio, ${\tilde{b}}_{bp}^m$, 13 values from 0.01 to 0.06 at an interval of 0.005 as well as 0.013 and 0.0183 were used in this study.

2. Materials and Methods

2.1. Radiative Transfer Basic Theory

The RTE can predict underwater radiance distributions given the IOPs and boundary conditions of the water body. The HydroLight solves the integrodifferential RTE along with its boundary conditions based on the time-independent and one-dimensional (unpolarized, depth-dependent) light field in horizontally homogeneous water bodies using the invariant embedding method. Directions are specified via an x–y–z cartesian coordinate system using the nadir angle $\theta$ and azimuth angle $\varphi$ with the x–y plane parallel to the water surface, with the +x and +z axis directions representing downwind and downward, respectively. The vector $\widehat{\xi }$ denotes a unit vector pointing in the desired direction, defined as follows.

$$\widehat{\xi }={\widehat{e}}_x\sin \theta \cos \varphi +{\widehat{e}}_y\sin \theta \sin \varphi +{\widehat{e}}_z\cos \theta$$

(2)

where ${\widehat{e}}_x$, ${\widehat{e}}_y$, and ${\widehat{e}}_z$ are unit vectors in the x, y, and z directions, respectively. $\theta$ is the polar angle measured from the nadir direction ${\widehat{e}}_z$, and the azimuth angle $\varphi$ increases counterclockwise from ${\widehat{e}}_x$, with values in the ranges $0\le \theta \le \pi$ and $0\le \varphi <2\pi$, respectively. The standard form of the RTE for radiance $L(z,\widehat{\xi },\lambda )$ propagating in the water is [30].

$$\cos \theta \frac{dL(z,\widehat{\xi },\lambda )}{dz}=-c(z,\lambda )L(z,\widehat{\xi },\lambda )+\underset{\mathsf{\Xi }}{\overset{}{\int }}L(z,{\widehat{\xi }}^{\prime },\lambda )\beta (z,{\widehat{\xi }}^{\prime }\to \widehat{\xi },\lambda )d\mathsf{\Omega }({\widehat{\xi }}^{\prime })+S(z,\widehat{\xi },\lambda )$$

(3)

where $c(z,\lambda )$ is the beam attenuation coefficient at geometric depth z and wavelength $\lambda$. The collection of all directions $\widehat{\xi }$ is designated the unit sphere $\mathsf{\Xi }$, which comprises all $(\theta ,\varphi )$ values such that $0\le \theta \le \pi$ and $0\le \varphi <2\pi$. The volume scattering functions (VSFs) from the direction ${\widehat{\xi }}^{\prime }$ to the direction $\widehat{\xi }$ is denoted as $\beta (z,{\widehat{\xi }}^{\prime }\to \widehat{\xi },\lambda )$. The quantity $d\mathsf{\Omega }({\widehat{\xi }}^{\prime })$ is the differential solid angle surrounding ${\widehat{\xi }}^{\prime }$, and integration is conducted over the entire range of solid angles. Note that the solid angle measurement of the entire set of all directions is $\mathsf{\Omega }(\mathsf{\Xi })=4\pi sr$, as the area of a sphere is $4\pi r^2$ (solid angle $\mathsf{\Omega }\equiv \mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}/\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s} \mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}$). The effective source term S is considered to include bioluminescence and inelastic scattering. In this paper, bioluminescence was not included, but we considered various inelastic scattering processes, such as chlorophyll and CDOM fluorescence, as well as Raman scattering by water; these processes are particularly important at longer wavelengths.

The RTE (Equation (3)) expressed in terms of the dimensionless optical depth $\zeta$ as the relationship between $\zeta$ and the beam attenuation coefficient c(z), as well as the geometric depth z ($d\xi \equiv c(z)dz$) is

$$\cos \theta \frac{dL(\zeta ,\widehat{\xi },\lambda )}{d\zeta }=-L(\zeta ,\widehat{\xi },\lambda )+w_0(\zeta ,\lambda )\underset{\mathsf{\Xi }}{\overset{}{\int }}L(\zeta ,{\widehat{\xi }}^{\prime },\lambda )\tilde{\beta }(\zeta ,{\widehat{\xi }}^{\prime }\to \widehat{\xi },\lambda )d\mathsf{\Omega }({\widehat{\xi }}^{\prime })+\frac{1}{c(\zeta ,\lambda )}S(\zeta ,\widehat{\xi },\lambda )$$

(4)

where $\tilde{\beta }$ is the volume scattering phase function ($\tilde{\beta }=\beta /b$, where $\beta$ and $b$ are the VSFs and total scattering coefficient, respectively, with units of sr⁻¹). $w_0$ is the single-scattering albedo, $w_0=b/\mathrm{c}$. The beam attenuation coefficient c can be expressed as $c=a+b$, where $a$ is the absorption coefficient. Equation (4) shows all quantities as a function of $\zeta$. The total scattering coefficient b is given by [12].

$$b(\lambda )=\underset{\mathsf{\Xi }}{\overset{}{\int }}\beta (\mathsf{\psi },\lambda ) d\mathsf{\Omega }=2\pi \underset{0}{\overset{\pi }{\int }}\beta (\psi ,\lambda )\sin \mathsf{\psi }d\mathsf{\psi }$$

(5)

The backscattering coefficient is defined as [21].

$$b_b(\lambda )=2\pi \underset{\pi /2}{\overset{\pi }{\int }}\mathsf{\beta }(\mathsf{\psi },\mathsf{\lambda })\sin \mathsf{\psi }d\mathsf{\psi }$$

(6)

where $\mathsf{\psi }$ is the scattering angle, and $\mathsf{\Omega }$ represents solid angles.

The downward and upwelling irradiances can be derived from the radiance through integration. The spectral downward plane irradiance beneath the water surface $E_d(0^{-})$ is expressed as [3,31].

$$E_d(0^{-})=\underset{{\mathsf{\Xi }}_d}{\overset{}{\int }}L(\theta ,\varphi ,\lambda )|\cos \theta | d\mathsf{\Omega }=\underset{\varphi =0}{\overset{2\pi }{\int }}\underset{\theta =0}{\overset{\pi /2}{\int }}L(\theta ,\varphi ,\lambda )|\cos \theta |\sin \theta d\theta d\varphi$$

(7)

and the spectral upward irradiance at null depth $E_u(0^{-})$ as [3]

$$E_u(0^{-})=\underset{{\mathsf{\Xi }}_u}{\overset{}{\int }}L(\theta ,\varphi ,\lambda )|\cos \theta | d\mathsf{\Omega }=\underset{\varphi =0}{\overset{2\pi }{\int }}\underset{\theta =\pi /2}{\overset{\pi }{\int }}L(\theta ,\varphi ,\lambda )|\cos \theta |\sin \theta d\theta d\varphi$$

(8)

where ${\mathsf{\Xi }}_u$ and ${\mathsf{\Xi }}_d$ are the upward and downward hemispheres of direction, respectively.

Additionally, defined by

$${\mathsf{\Xi }}_u\equiv \mathrm{a}\mathrm{l}\mathrm{l} (\mathsf{\theta },\mathsf{\varphi }) \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \pi /2<\theta \le \pi \mathrm{a}\mathrm{n}\mathrm{d} 0\le \varphi <2\pi$$

$${\mathsf{\Xi }}_d\equiv \mathrm{a}\mathrm{l}\mathrm{l} (\mathsf{\theta },\mathsf{\varphi }) \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} 0<\theta \le \pi /2 \mathrm{a}\mathrm{n}\mathrm{d} 0\le \varphi <2\pi$$

2.2. Inherent Optical Properties Model

In radiative transfer simulations, IOPs (e.g., $a$, $b$, c, $\beta$, and $\tilde{\beta }$) are necessary input quantities used as geometric functions in equations and models. We assumed that the study area (Lake Nansihu) is a homogeneous water body in which IOPs do not change with depth. The IOP parameters were computed on the basis of the following models. The spectral total absorption coefficient $a(\lambda )$ was calculated as follows [32]:

$$a(\lambda )=a_w(\lambda )+a_{ph}(\lambda )+a_{NAP}(\lambda )+a_{CDOM}(\lambda )$$

(9)

where $a_w(\lambda )$, $a_{CDOM}(\lambda )$, $a_{ph}(\lambda )$, and $a_{NAP}(\lambda )$ are the absorption coefficients of pure water, CDOM, phytoplankton, and non-algal (mineral) particles, respectively. $a_w(\lambda )$ was obtained from Pope et al. (1997) [33]. The terms $a_{ph}(\lambda )$ and $a_{NAP}(\lambda )$ can be expressed as multiply normalized chlorophyll-specific or mass-specific absorption [34,35] for each chlorophyll or mineral particle concentration (${\mathrm{C}}_{ph}$ or ${\mathrm{C}}_{NAP}$). $a_{CDOM}$ was calculated as [36].

$$a_{CDOM}(\lambda )=a_{CDOM}(440)e^{-0.014(\lambda -440)}$$

(10)

Equation (9) can be expressed as

$$a(\lambda )=a_w(\lambda )+{a^{\mathrm{*}}}_{ph}(\lambda ){\mathrm{C}}_{ph}+{a^{\mathrm{*}}}_{NAP}(\lambda ){\mathrm{C}}_{NAP}+a_{CDOM}(440)e^{-0.014(\lambda -440)}$$

(11)

The spectral total scattering coefficient $b(\lambda )$ was calculated as follows [12]:

$$b(\lambda )=b_w(\lambda )+b_p(\lambda )$$

(12)

and the spectral particles scattering coefficient was determined as [37]:

$$b_p(\lambda )=0.407(\frac{660}{\lambda }){{\mathrm{C}}_p}^{0.795}$$

(13)

where ${\mathrm{C}}_p$ is the particle concentration, which is equal to ${\mathrm{C}}_{ph}$ plus ${\mathrm{C}}_{NAP}$ and VSFs is [29]:

$$\mathsf{\beta }(\mathsf{\psi },\mathsf{\lambda })={\mathsf{\beta }}_w(\mathsf{\psi },\mathsf{\lambda })+{\mathsf{\beta }}_p(\mathsf{\psi },\mathsf{\lambda })$$

(14)

The volume scattering phase function can be written as [38]:

$$\tilde{\mathsf{\beta }}(\mathsf{\psi },\mathsf{\lambda })=\frac{b_w(\lambda )}{b(\lambda )}{\tilde{\mathsf{\beta }}}_w(\mathsf{\psi })+\frac{b_p(\lambda )}{b(\lambda )}{\tilde{\mathsf{\beta }}}_p(\mathsf{\psi },\mathsf{\lambda })$$

(15)

The spectral total backscattering coefficient $b_b(\lambda )$ was calculated as follows [21]:

$$b_b(\lambda )=b_{bw}(\lambda )+b_{bp}(\lambda )$$

(16)

where $b_{bw}(\lambda )$ and $b_{bp}(\lambda )$ are the backscattering coefficients of pure water and total suspended particles (phytoplankton $b_{bp}^c$ and mineral particle $b_{bp}^m$), respectively. $b_w(\lambda )$ and ${\mathsf{\beta }}_w(\mathsf{\psi },\mathsf{\lambda })$ were obtained from Zhang et al. [39]. $b_{bw}(\lambda )$ is half of the total molecular scattering coefficient, $b_w(\lambda )$, and the backscattering ratio ${\tilde{b}}_{bw}(\lambda )=0.5$. The particle phase function with the ${\tilde{b}}_{bp}$ values from 0.0004 to 0.06 [29] were applied to the RTE in this study. For the chlorophyll-bearing particle ratio, ${\tilde{b}}_{bp}^c$, 15 values were selected, including 0.005 to 0.018 at an interval of 0.001 as well as 0.0024 and 0.0183. For the mineral particle ratio, ${\tilde{b}}_{bp}^m$, 12 values from 0.01 to 0.06 at an interval of 0.005 as well as 0.0183 were used.

The average number of collisions, $\stackrel{-}{n}$, is an intuitively efficient representation of diffuseness inside the radiant field, which is, in turn, directly related to the $f^{\prime }$ factor [12]. The $\stackrel{-}{n}$ experienced by photons before they escape from water can be expressed as: $\stackrel{-}{n}={(1-\stackrel{-}{w})}^{-1}$ or $\stackrel{-}{n}=1+(b/a)$, where $\stackrel{-}{w}$ is the single-scattering albedo, $\stackrel{-}{w}=b/(a+b)$, and a and b are the absorption and scattering coefficients, respectively. Notably, we used $\stackrel{-}{n}$ rather than $\stackrel{-}{w}$ because $\stackrel{-}{n}$ is unbounded and more intuitive than $\stackrel{-}{w}$ [13], especially when multiple scattering is dominant in inland waters.

2.3. Input Parameters

The input parameters for Lake Nansihu simulations included chlorophyll-a and particulate concentrations, as well as depth and solar zenith angle based on in situ data (

Table 1. Data were collected at 27 sites during a Lake Nansihu cruise in 2015.

Parameters	Min	Max	Mean	Std
Chla [mg m−3]	5.00	22.32	8.94	3.38
TSM [mg L−1]	1.02	31.21	9.17	7.01
Depth [m]	0.79	2.33	1.37	0.32
Solar zenith (°)	15.70	48.15	27.08	9.49

Min, Max, Mean, and Std stand for minimum, maximum, mean, and standard deviation shown here.

) collected in the region of the lake, namely, 34°33′29″–34°39′49″N and 117°11′12″–117°18′52″E. Particulate material (TSM) was collected on 25 mm Whatman GF/F glass fiber filters. Chlorophyll-a concentration (Chla) was collected in 47 mm GF/F glass fiber filters and measured [40]. Solar zenith was computed by the time in situ.

The radiative transfer simulations were conducted using the HydroLight 6.0 code with the following input parameters in addition to the parameters listed in Table 1:

• Wavelength, $\lambda$ (100 values, from 400 nm to 750 nm at an interval of 5 nm)
• Bottom reflectance (macrophytes and clean seagrass, Figure 1a)
• Chlorophyll-bearing particle ratio, ${\tilde{b}}_{bp}^c$ (16 values, from 0.005 to 0.018 at an interval of 0.001 as well as 0.0024 and the Petzold average of 0.0183)
• Mineral particle ratio, ${\tilde{b}}_{bp}^m$ (13 values, from 0.01 to 0.06 at an interval of 0.005 as well as 0.013 and the Petzold average of 0.0183).
• CDOM (0.30 m⁻¹ at 440 nm)
• Wind speed: 5 m/s
• Real index of refraction, n: 1.34
• Cloud coverage: 0%
• Airmass type: continental
• Relative humidity: 80.0%
• Aerosol optical thickness at 550 nm: 0.261
• Total ozone: 300.0 Dobson units

Figure 1. (a) Bottom irradiance reflectance spectra obtained from HydroLight; (b) Phase functions of particles with backscatter fractions from 0.0024 to 0.06 and pure water with scattering angle.

Figure 1. (a) Bottom irradiance reflectance spectra obtained from HydroLight; (b) Phase functions of particles with backscatter fractions from 0.0024 to 0.06 and pure water with scattering angle.

Two bottom types, the clean seagrass, and macrophyte, were used as provided with HydroLight 6.0 (Figure 1a). The average clean seagrass reflectance was based on measurements between 350 and 800 nm of 100% Thalassia measured by R. Zimmerman at Lee Stocking Island, Bahamas. Additionally, the Bottom reflectance spectrum for the average macrophyte is based on measurements between 400 and 750 nm taken near Lee Stocking Island, Bahamas, during the Coastal Benthic Optical Properties (CoBOP) experiment. Particle phase functions with Petzold average [29] and Fournier–Forand [18] values of ${\tilde{b}}_{bp}$ from 0.0024 to 0.06 were used for the numerical simulations (Figure 1b). The scattering phase function for pure water was presented by Zhang et al. [39].

In addition, normalized sky radiances were computed using the Harrison and Coombes normalized sky radiance model [41]. Diffuse and direct sky irradiances were computed using the RADTRANX model [42]. Chlorophyll fluorescence quantum efficiency was set to 0.020, and the CDOM absorption coefficient used for the fluorescence calculation was obtained from the CDOM component of the IOP model. Raman scattering was included in this run, with a Raman scattering coefficient of 2.60 × 10⁻⁴ m⁻¹ at the reference wavelength of 488 nm. The azimuthally isotropic Cox–Munk surface model was used for the surface simulation. The flow chart of $f^{\prime }$ factor calculation was shown in Figure 2. In this study, radiative transfer simulation was applied to calculate the ƒ′ values using the above IOPs models based on concentrations data of Chla, TSM, and CDOM combined with realistic boundary conditions, including sky radiance, sea state, bottom BRDF, and fluorescence. $f^{\prime }$ factor for 16 ${\tilde{b}}_{bp}^c$ values, 13 ${\tilde{b}}_{bp}^m$ values, and 2 bottom reflectance was calculated at 27 stations, and the total amount of $f^{\prime }$ factor was 11,232 (N = 11,232).

3. Results and Discussion

The irradiance reflectance that is just below the surface, ${R(0}^{-})$, was simulated with HydroLight. The value of $f^{\prime }$ factor equals the ratio between ${R(0}^{-})$ and $b_b/(a+b_b)$. The relationship between ${R(0}^{-})$ and $b_b/(a+b_b)$ is shown in Figure 3. The black solid dot represents our simulated values obtained using the radiative transfer model described above. We compared several classical $f^{\prime }$ models with our results (Figure 3), which is larger than the value of 0.33 reported by Morel and Prieur (1977) [1]. The green and grey regions on the graph showed the results derived from the formula of Kirk (1984) [2] and Morel et al. (2002) [6], respectively. It could be seen that Kirk and Morel’s methods included only partial values, and some of our values were larger than these methods computed.

These models primarily focused on ocean water, typically assumed to possess infinite depth, which was inapplicable to our research region (Figure 3). Our research region, Nansihu Lake, has a maximum depth of 2.33 m; the bottom sediment may affect the upwelling radiances and $f^{\prime }$ factor [43,44] so that the bottom reflectance must be considered in the radiative transfer model. Moreover, the composition and optical properties of inland waters are more complex than the ocean [45,46,47]; the maximum concentrations of chlorophyll-a and TSM are 22.32 mg m⁻³ and 31.21 mg L⁻¹, respectively, larger than the previous ocean studies [6,7,8]. Therefore, for inland waters such as Lake Nansihu, further studies, and models based on IOPs and solar zenith angle were needed [13,48].

The spectrum of $f$ and $f^{\prime }$ was shown in Figure 4. The $f$ and $f^{\prime }$ values showed similar trends to $f^{\prime }$ for 16 ${\tilde{b}}_{bp}^c$ values, 13 ${\tilde{b}}_{bp}^m$ values, and 2 bottom reflectances at 27 stations, a total of N = 11,232 (Figure 4). The coefficient $f^{\prime }$ should be used if $b_b$ is not small with respect to $a$ rather than the $f$ factor. In this study, the backscattering coefficient $b_b$ at some stations was not small with respect to the absorption coefficient $a$ due to the effect of particulate (Table 1); therefore, $f^{\prime }$ was used rather than $f$ [6,13]. The value $f^{\prime }$ factor was relatively small at 400–650 nm, around 0.4, while there was a large change in $f^{\prime }$ factor at 650–750 nm, with a maximum value approaching 18. Therefore, those situations were separated to study.

Three particle phase functions, ${\tilde{b}}_{bp}$ of 0.01, 0.0183, and 0.03 were used to simulate and calculate the $f^{\prime }$ factor in the 400–650 nm, which were often represented for the collective particle scattering research of mineral particles and phytoplankton [7,8,27,28,48,49]. At 400–650 nm, as the solar zenith angle increased, the value of the $f^{\prime }$ increased (Figure 5) for three values of ${\tilde{b}}_{bp}$ of 0.01, 0.0183 and 0.03. It can be seen that particulate backscattering ratios had little influence on $f^{\prime }$ factor and can be ignored in the 400–650 nm range. Hence, we proposed an approximation of $f^{\prime }$ that is independent of the water body and dependent only on the solar zenith angle, ${\theta }_s$. $f^{\prime }$ increases continuously as $(1-\cos {\theta }_s)$, rising from 0.33 to 0.43 in Figure 5. The dependence of $f^{\prime }$ on $(1-\cos {\theta }_s)$ is approximately linear, and linear regression analysis of the data yields the relationship

$$f^{\prime }=0.3328+0.2517(1-\cos {\theta }_s)$$

(17)

Combining this relationship with Equation (1), we obtain

$$R(0^{-})=(0.3328+0.2517(1-\cos {\theta }_s))\frac{b_b}{a+b_b}400-650\mathrm{n}\mathrm{m}$$

(18)

The coefficient of determination R² was 0.8929, and the sum of squares of errors and root mean square error were 0.0038 and 0.0071, respectively. The results indicated that the factor $f^{\prime }$ increased with ${\theta }_s$ in a rather regular way, when ${\theta }_s<{50}^{°}$; and was linearly related to when $\cos {\theta }_s>0.65$ (Figure 5). This model was similar to Kirk’s model and Morel’s approach in which $f$ or $f^{\prime }$ is a function of solar zenith angle.

Figure 5. $f^{\prime }$ factor plotted as a function of ($1-\cos {\theta }_s$) and the corresponding linear fit for Nansihu Lake with three particulate backscattering ratios (${\tilde{b}}_{bp}$) at 400–650 nm. R² = 0.8929, sum of squares of errors = 0.0038, root mean square error = 0.0071. Gray circle, blue triangle, and green square represent different values of ${\tilde{b}}_{bp}$ (0.01, 0.0183, and 0.03, respectively).

Figure 5. $f^{\prime }$ factor plotted as a function of ($1-\cos {\theta }_s$) and the corresponding linear fit for Nansihu Lake with three particulate backscattering ratios (${\tilde{b}}_{bp}$) at 400–650 nm. R² = 0.8929, sum of squares of errors = 0.0038, root mean square error = 0.0071. Gray circle, blue triangle, and green square represent different values of ${\tilde{b}}_{bp}$ (0.01, 0.0183, and 0.03, respectively).

In the 650–750 nm range, the coefficients of determination (R²) for $f^{\prime }$ with $\cos {\theta }_s$ and ${\tilde{b}}_{bp}$ are presented in Figure 6. The $f^{\prime }$ factor and ${\tilde{b}}_{bp}$ exhibit a large coefficient of determination (R² $\approx$ 0.8 at 660–710 nm), whereas $f^{\prime }$ is unrelated to $\cos {\theta }_s$ (R² $\approx$ 0.0), which showed no significant linear positive correlation. Hence, we assumed that, in the range of 650–750 nm, the variation of $f^{\prime }$ factor mainly depends on the IOPs ($\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$), and is independent of the solar zenith angle as it had little influence on the $f^{\prime }$ factor in the 650–750 nm (Figure 6).

The $f^{\prime }$ factor had a large change with different values of $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ (Figure 7), and decreased with the average number of collisions ($\stackrel{-}{n}$) increasing. Therefore, we propose an approximation of the $f^{\prime }$ factor in the range of 650–750 nm with $\stackrel{-}{n}$ values at 600 nm divided 27 stations into six types, provided that $f^{\prime }$ factor is independent of ${\theta }_s$ and dependent on ${\tilde{b}}_{bp}$. Aside from the impact of the average number of collisions, the $f^{\prime }$ factor was significantly influenced by the changes in the backscattering ratios since variation in the ${\tilde{b}}_{bp}$ results in changes VSFs, which ultimately affected the bidirectional reflectance distribution function [6,7]. In the 650–750 nm, $f^{\prime }$ factor mainly followed a Gaussian distribution, and backscattering ratios determined the height of the peak of the Gaussian function, which was not reported in previous studies since there are weakly remote-sensing reflectance in the red and near-infrared bands for the ocean waters [6,7,8,30,31]. The dependence of $f^{\prime }$ factor on ${\tilde{b}}_{bp}$ is an approximately Gaussian function and is expressed with the equation:

$$f^{\prime }=A{e}^{-{(\frac{\lambda -{\lambda }_0}{B})}^2}+C$$

(19)

where ${\lambda }_0=685\mathrm{n}\mathrm{m}$, B = 14.24 ± 0.12, C = 0.374 ± 0.016, and the coefficient was obtained from the $A$ lookup table according to the $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ values (

Table 2. Lookup table of mean A values and standard deviation as a function of $\stackrel{-}{n}$ at 600 nm and ${\tilde{b}}_{bp}$.

Parameter			${\stackrel{-}{\mathit{n}}}_{600}$
${\tilde{\mathit{b}}}_{\mathit{b}\mathit{p}}$	2.0–2.2	2.2–2.5	2.5–3.0	3.0–3.5	3.5–4.0	4.0–5.0
0.003–0.004	16.21 ± 0.25
0.004–0.005	12.87 ± 0.20	10.11 ± 0.16	6.21 ± 0.10
0.005–0.006	10.26 ± 0.16	8.09 ± 0.13	5.11 ± 0.08	4.39 ± 0.07	3.17 ± 0.05
0.006–0.007	9.00 ± 0.14	6.91 ± 0.11	4.66 ± 0.08	3.71 ± 0.06	2.51 ± 0.04
0.007–0.008	7.78 ± 0.12	6.05 ± 0.10	4.01 ± 0.07	3.45 ± 0.06	2.23 ± 0.04
0.008–0.009	6.86 ± 0.11	5.40 ± 0.09	3.66 ± 0.06	3.06 ± 0.05	1.88 ± 0.03
0.009–0.01	6.19 ± 0.10	4.76 ± 0.08	3.14 ± 0.06	2.61 ± 0.04	1.74 ± 0.03
0.01–0.012	5.35 ± 0.08	4.12 ± 0.07	2.66 ± 0.05	2.21 ± 0.04	1.52 ± 0.03	1.16 ± 0.02
0.012–0.014	4.53 ± 0.07	3.54 ± 0.06	2.30 ± 0.05	1.93 ± 0.03	1.36 ± 0.02	0.99 ± 0.02
0.014–0.016	3.92 ± 0.06	3.07 ± 0.05	2.03 ± 0.05	1.71 ± 0.03	1.22 ± 0.02	0.87 ± 0.02
0.016–0.018	3.47 ± 0.05	2.72 ± 0.04	1.82 ± 0.04	1.54 ± 0.03	1.03 ± 0.02	0.77 ± 0.01
0.018–0.02	3.13 ± 0.05	2.44 ± 0.04	1.55 ± 0.04	1.30 ± 0.02	0.85 ± 0.02	0.70 ± 0.01
0.02–0.025	2.78 ± 0.04	2.13 ± 0.03	1.27 ± 0.03	1.07 ± 0.02	0.72 ± 0.01	0.59 ± 0.01
0.025–0.03		1.78 ± 0.03	1.08 ± 0.03	0.92 ± 0.02	0.61 ± 0.01	0.49 ± 0.01
0.03–0.035			6.21 ± 0.10	0.81 ± 0.02	0.57 ± 0.01	0.42 ± 0.01
0.035–0.04				0.74 ± 0.01	3.17 ± 0.05	0.34 ± 0.01
0.04–0.045				4.39 ± 0.07	2.51 ± 0.04
0.045–0.055						0.29 ± 0.01

). The central Gaussian function was around 685 nm, which always had a maximum of $f^{\prime }$ factor. The full width at the half of the maximum was about 23.71 nm according to the calculation of the value of B, which had little effect of the $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ values in this study. The larger the particulate backscattering ratios, the smaller the $f^{\prime }$ values. Thus, we can express $R(0^{-})$ as an explicit function of $a$, $b_b$, and the coefficient $A$ in the 650–750 nm range:

$$R(0^{-})=\left(A{e}^{-{(\frac{\lambda -685}{14.24})}^2}+0.374\right)\frac{b_b}{a+b_b}650-750 \mathrm{n}\mathrm{m}$$

(20)

This research has shown that particle backscattering can alter the BRDF in the 650–750 nm, and $f^{\prime }$ values decreased as the backscattering increased. This conclusion was consistent with the findings of Loisel and Morel [13,37], which indicated an increase in backscattering leads to BRDF change from anisotropy to isotropy. However, we still do not know what the critical value of particle backscattering is. In other words, the BRDF’s impact on remote sensing must be considered in Nansihu Lake.

Mean $f^{\prime }$ values as a function f the solar angle, ${\tilde{b}}_{bp}$ and wavelength are proposed as Equations (17) and (19), and A values obtained using Equation (19) are provided in Table 2. Detailed lookup tables for other values, including B and C, are available upon request from the authors.

The value of A, B, and C in Equation (19) will affect the accuracy of the model’s estimation of $f^{\prime }$ factor, especially the value of A. The smaller the values of $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$, the greater the uncertainties brought by the value of A (Table 2). The maximum standard deviation was around 0.25 of the value of A when the $\stackrel{-}{n}$ was between 2.0 and 2.2 and the ${\tilde{b}}_{bp}$ was between 0.003 and 0.004, respectively. Root mean square error (RMSE) was used as an indicator of the uncertainty. The maximum error always was at 687.5 nm, as our model was ${\lambda }_0=685\mathrm{n}\mathrm{m},$ and in this band, it always had a maximum. At 678.5 nm, the maximum RMSE was 0.68 when the $\stackrel{-}{n}$ was between 2.0 and 2.2 and the ${\tilde{b}}_{bp}$ was between 0.003 and 0.004, respectively; and the minimum value was 0.01 when the $\stackrel{-}{n}$ was between 4.0 and 5.0 and the ${\tilde{b}}_{bp}$ was between 0.045 and 0.055, respectively. The RMSE showed that the smaller the values of $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$, the greater the uncertainties because the $f^{\prime }$ factor always had a maximum value at 687.5 nm and decreased with the values of $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ increasing in our results. This is an important finding that has not been mentioned in previous studies [6,7,8], which focused on less than 660 nm.

To the extent that $f^{\prime }$ values increase as A values increase, when $\stackrel{-}{n}$ increases, a multiple-scattering regime develops in which the upward field tends to become isotropic and less sensitive to ${\tilde{b}}_{bp}$ or VSFs, with lower $f^{\prime }$ values.

This study demonstrates that the ${\mathit{f}}^{\mathbf{\prime }}$ factor is a function of the solar zenith angle, $\stackrel{-}{\mathit{n}}$, and ${\tilde{\mathit{b}}}_{\mathit{b}\mathit{p}}$ at various wavelengths under the condition of ${\mathit{a}}_{\mathit{C}\mathit{D}\mathit{O}\mathit{M}(\mathbf{440} \mathbf{n}\mathbf{m})}\mathbf{=}\mathbf{0.3}$ and two different bottom reflectances. It is necessary to discuss the effect of ${\mathit{a}}_{\mathit{C}\mathit{D}\mathit{O}\mathit{M}}$ and two bottom reflectances on the study the ${\mathit{f}}^{\mathbf{\prime }}$ factor. We investigated whether the concentration of CDOM had an impact on the study. Figure 8 shows the ${\mathit{f}}^{\mathbf{\prime }}$ spectra at three stations (Station 10, 11, and 21) with ${\tilde{\mathit{b}}}_{\mathit{b}\mathit{p}}$ of 0.013, 0.01, and 0.0068, respectively. CDOM absorption ranged from 0.1 to 0.9 at an interval of 0.05 at 440 nm.

Analysis of variance (ANOVA) was used to determine the statistical significance of $f^{\prime }$ factor with the CDOM absorption and two bottom types, namely, seagrass and macrophytes; these reflectance spectra are shown in Figure 1a. The ANOVA showed that the impact of the magnitude of $a_{CDOM}$ at 440 nm on $f^{\prime }$ factor (

Table 3. ANOVA results for the effects of $f^{\prime }$ on the magnitude of $a_{CDOM}$ at 440 nm. SSR and MSR represent the sum of squares regression and mean square regression, respectively.

Station	n	$\mathbf{ANOVA} \mathbf{of} {\mathit{f}}^{\mathbf{\prime }}$$\mathbf{Factor} \mathbf{Differences} \mathbf{in} {\mathit{a}}_{\mathit{C}\mathit{D}\mathit{O}\mathit{M}}$ at 440 nm
		SSR	MSR	F-Value	p-Value
10	17	0.001	0.00005	0	1
11	17	0.005	0.00030	0	1
21	17	0.000	0.00019	0	1

) was not significantly different. The values of p were equal to 1 in the three stations, indicating that the effect of $a_{CDOM}$ on $f^{\prime }$ can be ignored when $a_{CDOM(440 \mathrm{n}\mathrm{m})}\le 0.9$. The effect of dissolved colored organic substance (CDOM) on the $f^{\prime }$ factor can be ignored if the absorption coefficient of CDOM ($a_{CDOM}$) is less than 0.9 at 440 nm; it was in agreement with the findings of Shen [49] in Taihu Lake. Hence, the dependence of the relationship between IOPs and AOPs on ${\tilde{b}}_{bp}$ was studied under this condition.

Likewise, the $f^{\prime }$ values obtained for clean seagrass and macrophyte bottom reflectances are displayed in Figure 9 for various ${\tilde{b}}_{bp}$ values. These had almost the same $f^{\prime }$ factor values across various ${\tilde{b}}_{bp}$ values under the same ${\tilde{b}}_{bp}$ condition.

The sum of squares regression (SSR) and mean square regression (MSR) were zero (Table 3), and the p value was equal to 1, demonstrating that the effect of two bottom reflectances was identical on the $f^{\prime }$ factor. However, bottom types and reflectance can affect the BRDF, and further research is neded to investigate [43].

This study shows that ƒ′ factor varied over the range of 0.33–16.64 in Lake Nansihu, a finite depth water, higher than the range observed for the ocean. The previous studies were mainly less than 660 nm [6,8,13], and the range observed for the ocean was 0.3–0.6; we had a good agreement within these ranges of previous studies, and the solar zenith is the main factor affecting the BRDF. Both of us described $f^{\prime }$ factor as a function of solar zenith angle and given a fitting function model. Moreover, we studied a wider range wavelength than the previous studies, 660–750 nm. we found that $f^{\prime }$ factor mainly followed a Gaussian distribution in the 650–750 nm, and the $f^{\prime }$ range of 0.33–16.64 in the 650–750 nm, which was less mentioned in previous studies. There was less research on the relationship between VSFs and BRDF, because of a lack of measurement of VSFs [7]. In our study, the VSFs was characterized by particulate backscattering ratios, which is more easily obtained. Our results indicated the larger the particulate backscattering ratios, the smaller the $f^{\prime }$ values. Additionally, the central of Gaussian function was around 685 nm, which always had a maximum of $f^{\prime }$ factor, and backscattering ratios determined the height of the peak of the Gaussian function. These findings and models can compensate for the shortcomings of $f^{\prime }$ factor research in the red and near infrared bands and improve the ability of modeling the irradiance distribution in inland waters. However, our studies were based on the chlorophyll-a and TSM concentrations ranging from 5.00 to 22.32 mg m⁻³ and from 1.02 to 31.21 mg L⁻¹, respectively. For inland turbid waters, TSM concentration is more higher than our study region [50,51,52], with TSM > 100 mg L⁻¹; the extremely turbid waters are perfectly diffuse scattering bodies, which leads to isotropy [13]. Hence, further research is needed to investigate the BRDF of turbid inland waters.

4. Conclusions

The ƒ′ factor of BRDF serves as a crucial connection between the IOPs and AOPs of inland waters. In this study, the radiative transfer model was applied to calculate the ƒ′ values, taking into account the finite depth. This study reveals that the behavior of photons in inland waters results from the combined effects of the solar angle, VSFs, and optically shallow bottoms. Two parametric models were proposed to evaluate ƒ′ values, with one as a function of solar angle in the 400–650 nm, and the other model of $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ and wavelength in the range of 650–750 nm. It was found that the ƒ′ has a continuous geometric increase as the value of $(1-\cos {\theta }_s)$ increases. However, it is relatively insensitive to solar angles in the 650–750 nm range. For this range, $f^{\prime }$ mainly follows a Gaussian function related to $\stackrel{-}{n}$ and ${\tilde{b}}_{bp}$ and wavelength with the reference wavelength ${\lambda }_0=685 \mathrm{n}\mathrm{m}$. In Lake Nansihu, the larger the scattering and backscattering, the smaller the ƒ′ factor values. In the future, measurement of in-water irradiance reflectance and IOPs will be necessary to validate the model and assessment of $f^{\prime }$ at various wavelengths in inland waters.