Monte Carlo simulation of spectral reflectance and BRDF of the bubble layer in the upper ocean

Lanxin Ma; Fuqiang Wang; Chengan Wang; Chengchao Wang; Jianyu Tan

doi:10.1364/OE.23.024274

1. Introduction

High wind speeds and breaking waves result in the continuous formation of a dense bubble layer near the sea surface that can extend to several meters. Breaking waves generate bubbles that serve an important function in numerous phenomena, including air−sea gas transfer [1], marine aerosol production [2, 3], and ambient noise generation [4]. In addition, as one of the integral constituents of seawater, bubbles enhance the total scattering and backscattering coefficients in the upper ocean, especially for a high-concentration of bubbles under high wind speed conditions [5]. These changes influence radiative transfer in both the ocean and atmosphere, which can have an important effect on the accuracy of ocean color remote sensing and aerosol optical thickness retrievals [6,7]. Focusing on these issues, the influence of bubble layer in the upper ocean on the remote sensing has been extensively studied in the past two decades. Zhang et al. [5] calculated the scattering and backscattering properties of bubble populations in the upper ocean using Mie theory, and found that the presence of bubbles enhances ocean reflectance, and shifts the ocean color towards green. The influence of submerged microbubbles on the diffuse asymptotic light field and remote sensing reflectance of the ocean color were studied by Flatau et al. [6]. Stramski et al. [7] estimated the effect of bubbles on remote sensing reflectance and underwater light field characteristics using the HydroLight numerical model. The effect of oceanic air bubbles on atmospheric correction of ocean color imagery was investigated numerically by Yan et al. [8]. The results showed that the effect is predominantly determined by the wavelength-dependent optical properties of the air bubbles. On the basis of experimental and theoretical research, Zhang et al. [9] investigated the spectral effect of bubbles injected by moving vessels on the remote-sensing reflectance of three different optical water types. Using a 3-D Monte Carlo model, Piskozub et al. [10] studied remote sensing reflectance, as well as the associated factors, including the optical properties of bubble clouds and the background optical properties of seawater. Christensen et al. [11] investigated the effect of ocean bubbles on satellite measurements using a linked HydroLight oceanic and 6S atmospheric radiative transfer model.

The upwelling water-leaving radiance field beneath the atmosphere–ocean interface and the emerging field are generally anisotropic. This anisotropy results from the anisotropic optical properties of the water body and the illumination conditions above the interface [12]. The Bidirectional Reflectance Distribution Function (BRDF) is a fundamental quantity for describing surface reflectance, and has been widely used to characterize the diffuse reflectance properties of oceanic water [12–14]. Morel and Gentili [12] studied the impact of chlorophyll concentration, view zenith angle, and solar zenith angle on the diffuse reflectance of oceanic water. Mobley et al. [13] investigated the effects of optically shallow bottoms on the upwelling radiance field beneath the ocean surface and remote sensing reflectance. Sayer et al. [14] presented a sea surface BRDF model that accounts for contributions to the observed reflectance from whitecaps, sun-glint, and underlight. Zhai et al. [15] studied the effects of the bio-optical model variations on the BRDF of oceanic water, improving to understanding of the uncertainty of ocean color remote sensing.

In the literature discussed above, the impact of bubble populations in the sea surface BRDF model is often neglected. Although the effects of bubble populations on remote sensing reflectance and the underwater light field have been extensively investigated [5–10], few studies have been conducted to investigate the influence of bubble populations on the BRDF of the bubble layer. In this study, the spectral reflectance and BRDF of the bubble layer are investigated using the Monte Carlo method. The radiative properties of bubbles are calculated using Mie theory. The inherent optical properties of seawater are related to four chlorophyll concentrations (Chls) of 0.01, 0.1,1, and 10 mg·m⁻³. The effects of wavelength, bubble coating, Chl, and bubble number density on the reflection characteristics of the bubble layer in the upper ocean are investigated.

2. Background

2.1 Size distribution of bubble populations

Bubbles generated immediately after the breaking of a wave are usually large, and are part of a rapidly changing bubble population. Small bubbles with a persistent population may remain tens of minutes after a wave breaking event [16,17]. The number density for bubbles frequently found in seawater with radii in the range of 10 μm to 150 μm varies from 10⁴ m⁻³ to 10⁷ m⁻³; the radii of bubbles can be as small as 0.01 μm, although these bubbles are usually difficult to measure [5,18]. Moreover, observations and modeling results show that bubbles are usually coated with an organic coating ranging in thickness from 0.01 μm for lipids to 1 μm for protein [18,19].

In this study, we focus on small persistent bubbles, including both clean and coated bubbles. Considering the variation of the bubble number density with wind speed and depth, the Hall–Novarini (HN) bubble population model [20,21] is used. The bubble size distribution n(r, z) (m⁻³ μm⁻¹) can be written as

n (r, z) = (1.6 \times 10^{4}) G (r, z) {(\frac{v_{10}}{13})}^{3} \exp [- \frac{z}{L (v_{10})}],

L (v) = {\begin{cases} 0.4 v_{10} \leq 7.5 \\ 0.4 + 0 .115 (v_{10} - 7.5) v_{10} > 7.5, \end{cases}

G (r, z) = {\begin{cases} {[r_{ref} (z) / r]}^{4} r_{min} \leq r \leq r_{ref} (z) \\ {[r_{ref} (z) / r]}^{4.37 + {(z/2 .55)}^{2}} r_{ref} (z) < r \leq r_{max}, \end{cases}

where z (m) denotes the underwater depth, r (μm) denotes the bubble radius, r_min and r_max are the minimum and maximum radii of the bubble population, respectively, r_ref (μm) is the reference radius, r_ref = 54.4 μm + 1.984 × 10⁻⁶z, and v₁₀ (m/s) is the wind speed measured at a height of 10 m. From Eq. (1), the bubble number density N(z) (m⁻³) can be calculated as

N (z) = \int_{r_{\min}}^{r_{\max}} n (r, z) d r ≃ (1.6 \times 10^{4}) \frac{r_{ref}^{4}}{3 r_{\min}^{3}} {(\frac{v_{10}}{13})}^{3} \exp [- \frac{z}{L (v_{10})}] .

The HN model assumes that no bubbles exist with a radius r smaller than 10 μm, so, in this study, the minimum radius r_min is set to 10 μm and bubbles with a smaller radius are neglected. The bubble number density varied with depth at different wind speeds are given in Fig. 1. As seen in this figure, although the maximum bubble number density for v₁₀ = 16.5 m/s can reach about 10⁸ m⁻³, which is higher than the previous measurement results, this value is considered reasonable here for the purpose of representing the full range of bubble number densities found in the sea [5,21]. Considering that the bubble number density at a depth of z = 8 m is sufficiently small in most cases, we choose the water body from the sea surface to z = 8 m as the target of this study. Moreover, in order to simplify the calculation, the water body is equally divided into eight layers with a depth interval of 1 m. The mean bubble number density

\bar{N}

in each layer is calculated as

{\bar{N}}_{i} = \int_{z_{1 i}}^{z_{2 i}} N (z) d z / (z_{2 i} - z_{1 i}), i = 1 \sim 8,

where z₁_i and z₂_i denote the upper and lower bound of layer i.

Fig. 1 Variations in the bubble number density with depth at different wind speeds.

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2.2 Radiative properties of bubble populations

The scattering coefficients of bubble populations b_bub(z) (m⁻¹) at depth z can be calculated as [5]

b_{b u b} (z) = 10^{- 12} N (z) {\bar{Q}}_{sca} \bar{S}, {\bar{Q}}_{sca} = 2.0,

with

\bar{S} = \int_{r_{\min}}^{r_{\max}} \frac{n (r, z)}{N (z)} π r^{2} d r = \int_{r_{\min}}^{r_{\max}} \frac{3 r_{\min}^{3}}{r_{ref}^{4}} G (r, z) π r^{2} d r ≃ 3 π r_{min}^{2},

where

{\bar{Q}}_{sca}

is the mean scattering efficiency, which equals 2.0 for bubbles with a mean radius

\bar{r} > 1

μm [5]. The mean scattering efficiency is equal to 1.0 when diffraction is ignored [22].

\bar{S}

(μm²) is the mean geometric cross-sectional areas of the bubble populations. As shown, the scattering coefficients of bubble populations b_bub depend mainly on the bubble number density N(z) and the minimum radius r_min. Figure 2 shows the scattering coefficients of bubble populations in each layer at different wind speeds.

Fig. 2 Scattering coefficients of bubble populations in each layer at different wind speeds.

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The phase function of bubble populations ${\tilde{β}}_{b u b} (z, θ)$ (sr⁻¹) at depth z is calculated as [23]

{\tilde{β}}_{b u b} (z, θ) = \frac{1}{b_{b u b}} \int_{r_{\min}}^{r_{\max}} Q_{β} (r, θ) π r^{2} n (r, z) d r,

where Q_β (r,θ) (sr⁻¹) is the scattering efficiency per unit solid angle in the direction θ for a single bubble with a radius r embedded in seawater, which is calculated using Mie theory [22]. The empirical relationship developed by Quan and Fry [24] is used to calculated the refractive index of seawater for a temperature of T = 25 °C and a salinity of S = 35 ppt. For coated bubbles, considering that the backscattering does not change much when the thickness of the coating exceeds 0.1 μm, we set the coating thickness to 0.1 μm in this study [5, 18]. Moreover, the component of the coating film is assumed to protein and the refractive index relative to water is set to 1.2 [18,23]. Figure 3 shows the scattering phase functions for both clean bubbles and coated bubbles at λ = 550 nm. The depth z is set to 1 m and the wind speed v₁₀ is set to 10 m/s. As shown, the effect of the bubble coating on backscattering is obvious. Moreover, the research of Stramski et al. [7] and Zhang et al. [23] indicated that phase functions of bubble populations (r > 10 μm) vary very little for different size distributions and wavelengths, such that the use of only these two phase functions for all depths and all wavelengths in the following analysis is reasonable.

Fig. 3 Scattering phase functions of coated bubbles and clean bubbles at λ = 550 nm.

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2.3 Inherent optical properties (IOP) of water bodies

In the simulation, pure seawater, chromophoric dissolved organic matter (CDOM), and phytoplankton can be considered as the optical components that constitute a certain background level of IOPs. The IOPs are derived from chlorophyll based bio-optical models for Chls of 0.01, 0.1, 1, and 10 mg·m⁻³. The four different background IOPs represent water bodies that range from clear to turbid. In addition, the needed IOPs are the absorption coefficient a_t (m⁻¹), scattering coefficient b_t (m⁻¹) and scattering phase function $\tilde{β} (θ)$ (sr⁻¹). The total spectral absorption coefficient of the water body a_t(λ) is the sum of the absorption coefficients of each absorption component

a_{t} (λ) = a_{w} (λ) + a_{p h} (λ) + a_{CDOM} (λ) .

where the absorption coefficient of pure seawater a_w(λ) is obtained from Krik [25]. The absorption coefficient of phytoplankton a_ph(λ) is equal to Chl multiplied by α_ph (m²·mg⁻¹), where α_ph is the specific absorption coefficient of phytoplankton, as indicated by Shooter et al. [26]. The absorption coefficient of CDOM a_CDOM(λ) is modeled as an exponential function of the light wavelength [27]

a_{CDOM} (λ) = a_{CDOM} (440) e^{- 0.014 (λ - 440)},

and a_CDOM(440) = 0.2[a_w(440) + 0.06Chl^0.65], where a_CDOM(440) and a_w(440) are the absorption coefficients of CDOM and water at λ = 440 nm, respectively.

The scattering coefficient of the CDOM is assumed to be negligible. Thus, the total spectral scattering coefficient of the water body b_t (m⁻¹) is the sum of the scattering coefficients of pure seawater b_w [28,29] and phytoplankton particles b_ph [30]

b_{t} (λ) = b_{w} (λ) + b_{p h} (λ) = 0.00193 {(\frac{550}{λ})}^{4.32} + (\frac{550}{λ}) 0.30 {Chl}^{0.62} .

Figure 4 shows the total spectral absorption coefficients and total spectral scattering coefficients of the water body as a function of wavelength for Chls of 0.01, 0.1, 1, and 10 mg·m⁻³.

Fig. 4 (a) Total spectral absorption coefficients and (b) total spectral scattering coefficients of the water body as a function of wavelength.

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The Petzold average-particle phase function is chosen to be the phase function of phytoplankton particles ${\tilde{β}}_{p h} (θ)$ (sr⁻¹) [29,31]. Based on recent studies on scattering of seawater by Zhang and Hu [32], the volume scattering function β_w(θ) (m⁻¹ sr⁻¹) is calculated as

β_{w} (θ) = β_{w} (90 °) (1 + 0.925 \cos^{2} θ),

where β_w(90°) is the volume scattering function at θ = 90°.

3. Monte Carlo modeling and code validation

Monte Carlo method is the most general and flexible technique for numerically solving equations of radiative transfer [33, 34]. Various researchers in the atmosphere and ocean optics community have been using this method extensively over the past several decades [10, 35,36]. In this study, the bubble layer in the upper ocean is modeled as a multilayer slab. An infinitely thin light beam is perpendicularly incident on the air−sea surface by default. After interacting with the bubble layer, the reflected photons are collected using the detectors, which are positioned in the hemisphere space, as shown in Fig. 5. Finally, the spectral reflectance R and BRDF of the bubble layer are calculated as [29, 37]

R = \frac{E_{u}}{E_{d}} = \frac{\sum_{2 π} M_{r} (Ω_{r})}{M_{0}},

BRDF(θ_{i}, ϕ_{i}, θ_{r}, ϕ_{r}) = \frac{d L_{r} (θ_{r}, ϕ_{r})}{L_{i} (θ_{i}, ϕ_{i}) \cos θ_{i} d Ω_{i}} ≃ \frac{L_{r} (θ_{r}, ϕ_{r})}{E_{d}} = \frac{M_{r} (Ω_{r})}{M_{0} Ω_{r} \cos θ_{r}},

where θ_i, ϕ_i define the solar zenith angle and solar azimuth angle, θ_r, ϕ_r define the view zenith angle and view azimuth angle, Δϕ = ϕ_i – ϕ_r defines the azimuth difference between the incident plane and view plane. Ω_i, Ω_r define the solid angle of the incident and reflected light, E_u and E_d define the upward and downward plane irradiance, L_i(θ_i, ϕ_i) and L_r(θ_r, ϕ_r) define the radiance of the incident and reflected light, M₀ is the total number of the photons that are incident on the air−sea surface, and M_r (Ω_r) is the number of photons that are collected with the use of detectors positioned in the hemisphere space outside the upper surface in the solid angle Ω_r. Note that the spectral reflectance R used in our study differs from that commonly adopted in ocean optics, both E_u and E_d are defined just above the air−sea surface.

Fig. 5 Schematic diagram of (a) light transfer in the bubble layer and (b) coordinates system.

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3.1 Monte Carlo rules

Monte Carlo theory for radiative transfer simulations has been extensively discussed in the literatures [38, 39]. Thus, this theory is only summarized in the following subsections.

1. Ray path length

The geometric distance l (m) is calculated based on a random number ξ between 0 and 1, l = lnξ/c_t, where c_t (m⁻¹) is the total attenuation coefficient in the bubble layer, and c_t = a_t + b_t + b_bub.

2. Scattering or absorption

Once the photon has moved, we determine whether the photon is to be scattered or absorbed by generating a random number ξ and comparing this number with the single scattering albedo ω which is defined as ω = (b_t + b_bub)/c_t. If ξ ≤ ω, the photon is scattered; otherwise the photon is absorbed. In addition, a new random number ξ₁, in conjunction with two variables η₁ and η₂, is generated to determine whether the photon is to be scattered by bubbles, phytoplankton particles or pure seawater. The two variables η₁ and η₂ are defined as

η_{1} = \frac{b_{b u b}}{b_{b u b} + b_{p h} + b_{w}},

η_{2} = \frac{b_{b u b} + b_{p h}}{b_{b u b} + b_{p h} + b_{w}} .

If ξ₁ ≤ η₁, the photon is scattered by bubbles; if η₁ < ξ₁ ≤ η₂, it is scattered by phytoplankton particles; otherwise, it is scattered by pure seawater.

3. Scattering distribution

From Eq. (12), we can conclude that the scattering phase function of pure seawater ${\tilde{β}}_{w} (θ)$ is of the form ${\tilde{β}}_{w} (θ)$ ∝1 + 0.925cos²θ. Then the cumulative distribution function of pure seawater P_w(θ) can be obtained by normalizing and integrating the scattering phase function [38]

P_{w} (θ) = - 0.118 \cos^{3} θ - 0.382 \cos θ + 0.5, 0 \leq P_{w} (θ) \leq 1.

For bubbles and phytoplankton particles, the scattering phase function satisfies the normalization condition [23, 38]

2 π \int_{0}^{π} \tilde{β} (θ) \sin (θ) d θ = 1,

then probability density function p(θ) for scattering angle θ can be obtained

p (θ) = 2 π \tilde{β} (θ) \sin (θ), 0 \leq θ \leq π .

To determine the value of scattering angle θ for a particular Monte Carlo realization, we set a random number ξ equal to the cumulative distribution function P(θ) [38]

P (θ) = \int_{0}^{θ} p (θ^{'}) d θ^{'} = 2 π \int_{0}^{θ} \tilde{β} (θ^{'}) \sin (θ^{'}) d θ^{'} = ξ, 0 \leq P (θ) \leq 1,

and solve for θ. Then we can get the relationship between random number ξ and the scattering angle θ as shown in Fig. 6.

Fig. 6 Determination of the scattering angle from random number for clean bubbles, coated bubbles, phytoplankton particles, and pure seawater.

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4. Air−sea interface

When light is incident on the air−sea interface, a random number ξ is generated to decide whether the light is reflected or refracted. The reflectance ρ for unpolarized light is given by [29]

ρ (θ_{i}, θ_{t}) = {\begin{cases} \frac{1}{2} {{[\frac{\sin (θ_{i} - θ_{t})}{\sin (θ_{i} + θ_{t})}]}^{2} + {[\frac{\tan (θ_{i} - θ_{t})}{\tan (θ_{i} + θ_{t})}]}^{2}}, θ_{i} \neq 0 \\ {(\frac{n_{i} - n_{t}}{n_{i} + n_{t}})}^{2}, θ_{i} = 0 \end{cases}

In Eq. (21), θ_i and θ_t are the incident angle and refracted angle, n_i and n_t are the refractive indexes for the incident side and for the transmitted side of the interface. If ξ ≤ ρ, the photon is reflected; otherwise the photon is refracted. Note also that when photons strike the air-sea interface from the water side, a critical incident angle θ_c above which there is 100% reflection has to be determined, θ_c = sin⁻¹(n_t /n_i). If θ_i ≥ θ_c, ρ = 1, the photon is reflected.

Moreover, the air−sea interface is a complicated rough surface related to wind speed, especially for high wind speed conditions [40]. For simplicity, the air−sea interface is regarded as a flat surface in this study. In the following discussion, wind speeds are just used to characterize different bubble number densities.

3.2 Code Validation

To validate the Monte Carlo code, we compared two model-simulated results with the results from other researchers.

1. BRDF·cosθ_r distribution with the light incident vertically

We use Monte Carlo code to compute for the BRDF·cosθ_r of a turbid medium slab with the following optical properties: thickness of d = 0.02 cm, refractive index relative to air of n = 1, scattering coefficient of b = 90 cm⁻¹, absorption coefficient of a = 10 cm⁻¹, and asymmetry factor of g = 0.75. In the simulation, the light incident that is vertically on the upper surface used 10⁷ photons. The results are compared with the data from van de Hulst’s Table [41], as shown in Fig. 7.

Fig. 7 BRDF·cosθ_r distribution as a function of view zenith angle.

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2. Light incident on the surface with an oblique incident angle

The BRDF of a semi-transparent coating layer with the following optical properties is calculated: optical thickness of τ = 0.1, refractive index relative to air of n = 1, and the bottom of the coating layer is a lambertian surface with reflectivity ρ = 0.8. In the simulation, the light incident on the upper surface has the solar zenith angles θ = 5°, 65°, and 85°, respectively. A total of 10⁷ photons are used. The results are compared with the data from Xia et al. [42], as shown in Fig. 8.

Fig. 8 BRDF distribution at different incident angles.

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As shown in Figs. 7 and 8, the results calculated using the code in this study compared reasonably well with the published results.

4. Results and discussion

4.1 Influence of bubble populations on spectral reflectance

Figure 9 illustrates the spectral reflectance R of the bubble layer for solar zenith angle θ_i = 0° with four different Chls at wavelengths ranging from 300 nm to 800 nm. Each graph for a given Chl includes nine reflectance curves as the simulations were made for both clean bubbles and coated bubbles with bubble number densities under four different wind speeds, and including a case for seawater with no bubbles. Given that the scattering of bubbles is unrelated to direct specular reflection of the upper surface, only the diffuse reflectance is considered in this study. As shown in this figure, light scattering by coated bubbles at high bubble number densities causes a significant enhancement of R across almost the entire spectrum when Chl is less than 1.0 mg·m⁻³. The effect of coated bubbles on R is more obvious than clean bubbles, which verifies the strong backscattering effect of coated bubbles. R decreases gradually with increasing wavelength for Chl less than 1.0 mg·m⁻³. However, a different trend is found in turbid waters when Chl = 10.0 mg·m⁻³. This is due to the variation in the IOP of the water body with wavelength, especially for the absorption coefficient. The strong scattering and weak absorption properties of water body enhance light scattering and lead to strong reflection. As shown in Fig. 4(a), there are two minimum values at λ = 370 nm and λ = 570 nm that correspond to the two peaks of R in Fig. 9(d). Meanwhile, it can be seen that the effect of bubbles on R when Chl = 10.0 mg·m⁻³ is less significant than in clear waters.

Fig. 9 Spectral reflectance of the bubble layer for solar zenith angle θ_i = 0° with Chls = 0.01, 0.1, 1.0, 10.0 mg·m⁻³. N_1~4 corresponds to the bubble number density at wind speed v₁₀ = 6, 10, 14 and 16.5 m/s, respectively.

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The ratio of spectral diffuse reflectance of the bubble layer under various bubble number densities to that for the case with no bubbles is presented in Fig. 10. In turbid waters when Chl = 10.0 mg·m⁻³, the ratio shows relatively little variation with wavelength even under high number density conditions. This pattern is observed for both clean and coated bubbles. For example, when v₁₀ = 16.5 m/s, the ratio for coated bubbles increases from about 1.08 to 1.32 with increasing wavelength across the entire spectrum. Thus we can conclude that in turbid waters bubble populations have a weak spectral effect on R, although the enhancement of R in the red spectral region can be somewhat greater than at the short-wavelength of the spectrum. As discussed in Ref [9], the reason lies in the fact that high concentrations of particulates in turbid waters increase the backscattering spectrum in a more spectrally neutral way, thus the effect of bubbles decreases obviously. As water becomes clearer, the effect of bubble populations on enhanced reflectance gradually increases, especially in the long-wavelength region. This ratio shows a similar varying tendency with the ratio of remote sensing reflectance calculated by Stramski et al. [7]. Meanwhile, the effect of coated bubbles on enhanced reflectance is much more significant than for clean bubbles. This result verifies the strong backscattering effect of coated bubbles in the backscattering process as well. For Chl = 0.01 mg·m⁻³ and v₁₀ = 16.5 m/s, the ratio in Fig. 10(a) is about 1.4 at λ = 300 nm both for coated and clean bubbles, while the maximum enhancement factor is about 11.5 and 3.0 (for coated and clean bubbles, respectively) in the long-wavelength region around λ = 800 nm. This result indicates that the sea surface of the bubble layer will appear greener or more yellowish than seawater with no bubbles, which is consistent with the previous results [7, 9].

Fig. 10 Ratio of the spectral reflectance under four wind speed conditions (v₁₀ = 6, 10, 14, 16.5 m/s) to that for the case with no bubbles in seawater. N_1~4 corresponds to the bubble number density at wind speed v₁₀ = 6, 10, 14 and 16.5 m/s, respectively.

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4.2 Influence of bubble populations on the BRDF·cosθ_r distribution for normal incidence

The presence of bubble populations increases the backscattering process for light in the bubble layer, which enhances the diffuse reflectance. Figure 11 illustrates the BRDF·cosθ_r distribution of the bubble layer at λ = 550 nm for solar zenith angle θ_i = 0° with four different Chls. As for Fig. 10, each graph for a given Chl includes nine curves. Also, given that the scattering of bubbles is unrelated to direct specular reflection of the upper surface, only the diffuse reflection is considered for BRDF·cosθ_r. As shown in Fig. 11, a strong enhancement of BRDF·cosθ_r can be observed with an increase in the bubble number density. As for the effect of bubble populations on R, coated bubbles have a greater impact on BRDF·cosθ_r than clean bubbles. As water becomes turbid, the value of BRDF·cosθ_r shows an increasing trend due to the scattering from phytoplankton particles. For example, in clear waters when Chl = 0.01 mg·m⁻³, the maximum BRDF·cosθ_r is about 0.0015 sr⁻¹. As Chl increases to 10.0 mg·m⁻³, the maximum BRDF·cosθ_r increases to about 0.0085 sr⁻¹.

Fig. 11 BRDF·cosθ_r of the bubble layer at λ = 550 nm for solar zenith angle θ_i = 0° and Chls = 0.01, 0.1, 1.0, and 10.0 mg·m⁻³. N_1~4 corresponds to the bubble number density at wind speed v₁₀ = 6, 10, 14 and 16.5 m/s, respectively.

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To improve the understanding of the effect of bubble populations on BRDF·cosθ_r at different Chls, the ratio of BRDF·cosθ_r under various bubble number densities to that for the case with no bubbles in seawater is presented as a function of Chl in Fig. 12. Note that this ratio does not change significantly for different view zenith angles. As shown in this figure, the ratio for both clean bubbles and coated bubbles is inversely proportional to exponentially increasing Chl. For coated bubbles, as Chl increases from 0.01 mg·m⁻³ to 10.0 mg·m⁻³, the maximum ratio decreases from 2.27 to 1.14. This trend arises because the scattering coefficient of phytoplankton particles increases significantly with Chl, whereas the scattering coefficient of the bubbles remains unchanged. Furthermore, it can be also observed that, the effect of bubbles on enhanced BRDF·cosθ_r increases significantly with increasing number density. This is because the scattering coefficient of the bubbles increases exponentially with wind speed, as shown in Fig. 2. Thus, we can conclude that the presence of bubble populations has a more significant influence on BRDF·cosθ_r in clear waters and under high bubble number densities.

Fig. 12 Ratio of BRDF·cosθ_r under four wind speed conditions (v₁₀ = 6, 10, 14, 16.5 m/s) to that for the case with no bubbles in seawater. N_1~4 corresponds to the bubble number density at wind speed v₁₀ = 6, 10, 14 and 16.5 m/s, respectively.

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4.3 Influence of bubble populations on the BRDF·cosθ_r distribution for oblique incidence

Figure 13 illustrates the BRDF·cosθ_r distribution in the incident plane in clear waters with Chl = 0.1 mg·m⁻³ at λ = 550 nm for solar zenith angle of θ_i = 0°, 30°, 45° and 60°. Three bubble number densities for coated bubbles at v₁₀ = 10, 14, and 16.5 m/s, and one case with no bubbles are examined. As for previous analyses, only diffuse reflection is considered. As shown in this figure, a specific reflecting characteristic can be observed in the case of oblique incidence, that is, the reflected energy in the backward direction, Δϕ = 0°, is greater than that in the forward direction, Δϕ = 180°, for seawater with no bubbles and under low bubble number densities. Nevertheless, as the bubble number density increases, the reflected energy distribution at Δϕ = 0° and Δϕ = 180° gradually changes. In clear waters when Chl = 0.1 mg·m⁻³, due to the low concentration and weak backscattering property of phytoplankton particles, seawater plays a dominant role in the total backscattering. According to the volume scattering function of seawater [32], forward and backward scattering have a much higher probability than scattering to the sides, i.e. 90°, so more scattering energy can be detected in the backward direction, Δϕ = 0°, for oblique incidence, as shown in Fig. 13(a). As the bubble number density increases, Mie scattering phase function of bubble populations will gradually play a leading role in the scattering process. From the phase function of bubble populations in Fig. 3, the scattering probability at scattering angle of 90°~170° is basically the same, and a peak at 180° can be found. Therefore, for oblique incidence, upwelling radiances scattering out of the sea surface are not symmetric any more after single scattering and more scattering energy can be detected in the backward direction, Δϕ = 0°. Furthermore, as shown in Fig. 3, the scattering intensity of the bubble populations in the backward is relatively small compared with the forward intensity, and there is a prominent broad peak at angles from 60° to 80°. After double scattering, the light whose first scattering angle is between 60° and 80° are more likely to scatter out through the sea surface in the forward direction, Δϕ = 180°, especially for larger incident angles. For example, when v₁₀ = 16.5 m/s, the corresponding zenith angle for the maximum BRDF·cosθ_r is approximately 18° at Δϕ = 0° when θ_i = 30°, whereas it gradually changes to approximately 24° at Δϕ = 180°when θ_i = 60°.

Fig. 13 BRDF·cosθ_r distribution in the incident plane in clear waters with Chl = 0.1 mg·m⁻³ at λ = 550 nm for solar zenith angle of θ_i = 0°, 30°, 45° and 60°. N_1~4 corresponds to the bubble number density at wind speed v₁₀ = 6, 10, 14 and 16.5 m/s, respectively.

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BRDF·cosθ_r distributions with azimuth angles ranging from 0° to 180° are presented in Fig. 14. Compared with seawater with no bubbles, the bubble populations have less effect on the BRDF·cosθ_r when v₁₀ = 10 m/s. As the wind speed increases to 14 m/s or 16.5 m/s, a significant enhancement of BRDF·cosθ_r can be observed almost in all directions. Thus, from Figs. 13 and 14, we can conclude that the presence of bubble populations under high bubble number densities will change not only the value of BRDF·cosθ_r but also the distribution.

Fig. 14 BRDF·cosθ_r distribution in clear waters with Chl = 0.1 mg·m⁻³ at λ = 550 nm for solar zenith angle of θ_i = 0°, 30°, 45° and 60°. The solar azimuth angle is ϕ_i = 180° and the view azimuth angle ranges from ϕ_r = 0° to 180°. N_1~4 corresponds to the bubble number density at wind speed v₁₀ = 6, 10, 14 and 16.5 m/s, respectively.

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5. Conclusion and further work

In this study, the spectral reflectance and BRDF of the bubble layer in the upper ocean are investigated using the Monte Carlo method. The HN bubble population model which considers the effect of wind speed and depth on the bubble size distribution is used. The radiative properties of bubble populations are calculated using Mie theory, and the IOPs of water bodies are related to four Chls of 0.01, 0.1, 1.0 and 10 mg·m⁻³. The effects of bubble number density, wavelength, bubble coating, and solar zenith angle on the reflection characteristics of bubble layers are investigated.

The results show that bubble populations under high wind speed conditions have a significant impact on the reflection characteristics of the bubble layer. This impact is mainly due to the strong backscattering of bubbles that are coated with an organic film. Compared with seawater with no bubbles, the spectral diffuse reflectance and diffuse BRDF·cosθ_r values in case of normal incidence can increase to 11.5 and 2.27 times in clear waters when Chl = 0.01 mg·m⁻³. For low wind speed conditions, the influence of bubble populations can be ignored. As water becomes turbid, the effect of bubble populations on the enhanced reflectance gradually decreases. Furthermore, the presence of bubble populations under high wind speed conditions changes not only the value of BRDF·cosθ_r, but also the distribution.

Given that bubbles smaller than 10 μm are ignored in the HN bubble population model, the influence of these smaller bubbles on the reflection characteristics of bubble layers is still an ongoing research issue. Nevertheless, the results presented in this study provide a quantitative assessment of the effect of bubbles in the sea surface BRDF model.

Moreover, because of the wave shadow effect, multiple scattering, and the uncertainty of the foam structure in the ocean surface layer, the influences of rough sea-surfaces are neglected in the present study. In future studies, the bubble layer in the upper ocean combined with the wind-blown surface and foam layer will be investigated.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 51336002, 51406042), the New Century Excellent Researcher Award Program from Ministry of Education of China (No. NCET-12-0152), and the National Natural Science Foundation of Shandong (No. ZR2014EEP001). A very special acknowledgment is made to the editors and referees whose constructive criticism has improved this paper.

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Monte Carlo simulation of spectral reflectance and BRDF of the bubble layer in the upper ocean

Abstract

1. Introduction

2. Background

2.1 Size distribution of bubble populations

2.2 Radiative properties of bubble populations

2.3 Inherent optical properties (IOP) of water bodies