Numerically modelling the reflectance of a rough surface covered with diesel fuel based on bidirectional reflectance distribution function

Ying Li; Shuang Dong; Qinglai Yu; Ming Xie; Zhichen Liu; Zhanjun Ma

doi:10.1364/OE.443178

1. Introduction

The development of maritime transportation and ocean engineering equipment promotes the scale of human-related marine activities. Meanwhile, inappropriate operations and accidents during oil exploration, extraction, refining, and transportation may result in oil spills, which cause a wide range of ocean pollution and endanger aquatic organisms [1]. Oil spill detection is of great significance to the emergency decision-making and marine ecological environment protection.

Since 1960s, remote sensing technology has been applied to monitor oceanic environment and developed in many forms (e.g., hyperspectral remote sensing, thermal infrared, laser-induced fluorescence, and synthetic aperture radar, etc.). Fingas and Brown have comprehensively discussed the applications of various types of remote sensing technology for oil spill monitoring [2,3]. The optical properties of the oil-polluted sea surface are different from those of the clean seawater surface [4], and therefore, provide a theoretical basis for oil spill detection. Clark et al. developed an imaging spectroscopy system called “Tetracorder” [5], which were applied to measure and analyze the reflection spectra of some typical petroleum hydrocarbons (PHCs) [6]. Liu et al. studied the differences of spectral characteristics between oil films and seawater, and classified oil films in the hyperspectral images [7]. Otremba et al. built a statistical model using Monte Carlo simulation, based on which they studied the changes in the reflectance of oil surface in visible band and its optical contrast against seawater under windy conditions [8,9].

Bidirectional reflectance distribution function (BRDF), which describes the distribution of the radiance in the outgoing direction (usually in a hemi-sphere, this will be discussed in details in the next section), is often introduced to discuss the reflectance of a rough surface, since it can provide additional information than Fresnel reflection and characterize the optical properties of the rough surface more comprehensively [10]. BRDF has been applied in the scientific studies and engineering applications on oceanography and related fields. Based on the Fresnel reflection and Fraunhofer method, Davies [11] proposed an empirical physical model of reflection and scattering properties that is suitable for rough sea surface conditions.

In this study, the advantages and limitations of the Fresnel reflection and BRDF on simulating randomly irregular micro-surfaces are studied. By combining electromagnetic theory with BRDF, a numerical model for oil spill detection in the case of rough sea surface is established. Reflectance spectra of clean and oil-polluted seawater are collected through lab experiments and used to construct and validate the proposed model. In addition, the relationships between the reflectance of the oil film and other parameters (e.g., windspeed, incident light angle, and complex refractive index of the oil pollutants, etc.) were discussed. It is expected that the numerical model presented in this study could provide theoretical guidance for the rapid detection of oil spills on rough sea surface.

2. Related studies

2.1 Modelling the optical reflectance of oil surface

Modelling the optical reflectance of oil surface has been an important topic in the field of oil spill science. Both physical and statistical models have been proposed in previous studies. For examples, Lu et al. developed a physical model for stable oil films on seawater based on two-beam interference theory [12]. They linked the attenuation coefficient with slick thickness and proposed an inversion theory of oil thickness based on this model [13]. However, this model only works for stable oil films in laboratory condition and is not applicable for oil slicks on the rough sea surface. Ren et al. combined two-beam interference theory with BRDF and formed a more comprehensive model for oil slicks on water surface [14]. Chen et al. applied the similar method and developed a physical model for oil emulsion [15]. Otremba built a statistical model of the oil-polluted seawater by applying Monte Carlo simulation and tracking the trajectories of photons [16–19]. Although Monte Carlo method has achieved high simulation accuracy with appropriate input settings, its calculation could be time-consuming and lead to low computing efficiency. In actual oil spill event, the distribution of oil pollutants changes over time. Therefore, the computing process of oil spill detection often repeats for many times along with the constant drift of oil pollutants. Thus, the real-time simulation and inversion based on a high-performance numerical model of oil slick is necessary for the decision-making in the cases oil spill events.

2.2 Definition and applications of BRDF

BRDF is defined by Nicodemus [20] as the ratio of reflected radiance per spherical angle over the incident irradiance. Its mathematical representation can be written in differential form as shown in Eq. (1):

(1)$$\begin{aligned} \textrm{BRDF(}{\theta _i}\textrm{,}\,\,{\phi _i}\textrm{,}\,\,{\theta _r}\textrm{,}\,\,{\phi _r}\textrm{,}\,\,\lambda ) &= \frac{{d{L_r}({\theta _r},\,\,{\phi _r},\,\,\lambda )}}{{d{E_i}({\theta _i},\,\,{\phi _i},\,\lambda )}}\\ &= \,\frac{{d{L_r}({\theta _r},\,\,{\phi _r},\,\,\lambda )}}{{{L_i}({\theta _i},\,\,{\phi _i},\,\,\lambda )\cos {\theta _i}d{\omega _i}}}\\ &= \frac{{d{L_r}({\theta _r},\,\,{\phi _r},\,\,\lambda )}}{{{L_i}({\theta _i},\,\,{\phi _i},\,\,\lambda )\cos {\theta _i}\sin {\theta _i}d{\theta _i}d{\phi _i}d\lambda }} \end{aligned}$$

where θ_r and ϕ_r are the angular coordinates of the reflection light; θ_i and ϕ_i are the angular coordinates of the incident light; E_i is the incident irradiance; L_i is the incident radiance; dL_r is the reflected radiance in an elemental solid angle dω_i. The geometry of incident and reflected light beams using spherical coordinate system is shown in Fig. 1.

Fig. 1. Geometry of incident and reflected light beam using spherical coordinate system

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BRDF has been widely applied in various fields including computer graphics, ocean optics, remote sensing, etc. Wynn et al. provided the working knowledge of the concepts and basic mathematics that are necessary to appreciate BRDF, as well as some exposures to the terminologies when discussing BRDF and BRDF-based optical techniques [21]. Ross et al. established a high-precision BRDF model for an anisotropic sea surface based on the assumption that the slopes and heights of sea surface follow a Gaussian distribution [22]. Bruneton et al. further improved this model, and applied it to simulate waves with trochoids in computer graphics [23]. These successful applications of BRDF indicate its potentials for modelling oil surface reflectance.

2.3 Davies model of sea surface

Davies studied and summarized the relationships between the reflectance and the statistics of the electromagnetic radiation from the random rough sea surface [11]. This theory was developed to characterize the scattering of the radar waves from the rough water surface, but it is equally valid in the visible optical domain [24]. Davies theory assumes that the heights of the surface irregularities also follow a Gaussian distribution, and employ the Kirchhoff approximation to derive the expressions for the electromagnetic energy reflected from rough surfaces [25].

The object’s surface should have the following optical properties [22]: (1) The surface is perfectly conducting, and would have a specular reflectance of unity if it were perfectly smooth; (2) The distribution of the heights of the surface irregularities follows a Gaussian distribution; (3) The autocovariance function of the surface irregularities also follows a Gaussian distribution.

3. Methodology

3.1 Structure of the proposed model

Electromagnetic surface radiating theory of heterogeneous materials is applied to build the optical model of oil film with rough microfacets [26]. If the oil surface is illuminated with a parallel beam of the monochromatic light, the reflectance may be contributed by two components: (1) the former arises from specular reflection, and (2) the other is induced by diffusing reflection or scattering. As shown Fig. 2, E_S is the field intensity at the surface; n is the local surface normal vector. k₁ and k₂ are the incident wave vector and the scattered wave vectors respectively.

Fig. 2. Schematic diagram of electromagnetic wave scattering from oil films at rough sea surface

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The phase function of the proposed model is built based on the surface reflectance model for anisotropic rough sea surface proposed by Ross et al. [22], which assumes that the rough sea surface is made up of many optically flat microfacets, and the probability distribution functions (PDFs) of slopes and heights follow a Gaussian distribution. The first-order term of the heights-distribution is proposed by Cox and Munk [26], the PDF of surface height is defined as Eq. (2).

(2)$$p(h)\, = \frac{1}{{\sigma \sqrt {2\pi } }}\exp \left( { - \frac{{{h^2}}}{{2{\sigma^2}}}} \right)$$

where h is the potential heights of sea surface, σ is the root-mean-square (RMS) deviation of the surface from the mean sea level (MSL).

The PDF of slopes is a Gram–Charlier expansion that also follows a Gaussian distribution (Eq. (3)):

(3)$$p(\zeta |U\,)\, = \frac{1}{{2\pi {\sigma _x}{\sigma _y}}}\exp \left[ { - \frac{1}{2}\left( {\frac{{\zeta_x^2}}{{\sigma_x^2}} + \frac{{\zeta_y^2}}{{\sigma_y^2}}} \right)} \right]$$

where p is the slope PDF of the sea surface; ζ_x and ζ_y are the slope distribution in the directions of the upwind and crosswind; U is the windspeed while σ_x² and σ_y² are the upwind and crosswind variances. Through the observations on the sea surface roughness based on sun slitter photographs, the variances were found by Cox and Munk [26] to be given as Eq. (4):

(4)$$\sigma _x^2 = 0.00316\,U\, \pm \textrm{0}\textrm{.004,}\quad \sigma _\textrm{y}^2 = 0.003{\kern 1pt} + 0.00192\,U\, \pm 0.004.$$

Because the oil films on sea surface damp the waves, the geometry of water surface is usually “smoothed” after it is polluted by the oil. Cox and Munk [26] further developed the empirical variances of oil slicks on water surface using the mixture of 40% diesel oil, 40% olive oil, and 20% fish oil as Eq. (5):

(5)$$\sigma _x^2 = 0.005 + 0.00078\,U\, \pm 0.002,\quad \sigma _y^2 = 0.003 + 0.00084\,U\, \pm 0.002.$$

Compared with the roughness coefficients of clean sea surface shown as Eq. (4), those of the oil surface shown as Eq. (5) is obviously smaller, and is applied in the proposed model.

The surface profile ζ is normally-distributed and isotropic. This usual status occurs when the surface arises from a random process. The specular reflectance, which carries out both Helmholtz integral and Kirchhoff boundary approximation [27], can be yield in Eq. (6) as follow:

(6)$${\rho _\lambda }\textrm{ = }{\rho _{\lambda ,s}}\exp \left[ { - {{\left( {\frac{{4\pi \sigma \cos {\theta_i}}}{\lambda }} \right)}^2}} \right]$$

In the case of the near-normal incidence, this equation can be simplified as Eq. (7):

(7)$${\rho _\lambda }\textrm{ = }{\rho _{\lambda ,s}}\exp \left[ { - {{\left( {\frac{{4\pi \sigma }}{\lambda }} \right)}^2}} \right]$$

where ρ_λ and ρ_λ,s are the specular reflectance of a slightly rough surface, and a hypothetical perfectly-smooth surface, respectively.

On the other hand, the scattering can be obtained using the differential angle dθ. The expression is specifically given in Eq. (8):

(8)$${r_d}(\theta )d\theta = {\rho _{\lambda ,s}}2{\pi ^4}{\left( {\frac{a}{\lambda }} \right)^2}{\left( {\frac{\sigma }{\lambda }} \right)^2}{({\cos {\theta_\textrm{r}} + 1} )^4}\sin {\theta _r}\,\exp \left[ { - {{\left( {\frac{{\pi a\sin {\theta_r}}}{\lambda }} \right)}^2}} \right]d\theta$$

By introducing azimuth angle (ϕ_i, ϕ_r), the expression of scattering reflectance can be given as Eq. (9):

(9)$$\begin{aligned} {r_d}({\theta ,\phi } )d\theta d\phi &= {\rho _{\lambda ,s}}2{\pi ^4}{\left( {\frac{a}{\lambda }} \right)^2}{\left( {\frac{\sigma }{\lambda }} \right)^2}{({\cos {\theta_r} + \cos {\theta_i}} )^4}\sin {\theta _r}\\ &\quad \cdot \exp \left\{ { - {{\left( {\frac{{\pi a}}{\lambda }} \right)}^2}[{{{\sin }^2}{\theta_r} + {{\sin }^2}{\theta_i} + 2\sin {\theta_r}\sin {\theta_i}\cos ({{\phi_i} - {\phi_r}} )} ]} \right\}d\theta d\phi \end{aligned}$$

Combining the contributions of specular reflection and scattering, the completed expression for the measured reflectance f_r can be given as Eq. (10):

(10)$$\begin{aligned} {f_r}({\theta ,\phi ,\lambda } )&= {\rho _{\lambda ,s}}\exp \left[ { - {{\left( {\frac{{4\pi \sigma }}{\lambda }} \right)}^2}} \right] + {\rho _{\lambda ,s}}2{\pi ^4}{\left( {\frac{a}{\lambda }} \right)^2}{\left( {\frac{\sigma }{\lambda }} \right)^2}{({\cos {\theta_r} + \cos {\theta_i}} )^4}\sin {\theta _r}\\ &\quad \cdot \exp \left\{ { - {{\left( {\frac{{\pi a}}{\lambda }} \right)}^2}[{{{\sin }^2}{\theta_r} + {{\sin }^2}{\theta_i} + 2\sin {\theta_r}\sin {\theta_i}\cos ({{\phi_i} - {\phi_r}} )} ]} \right\}d\theta d\phi \end{aligned}$$

Applying Kirchhoff approximation [27] into Eq. (10), the expression of the proposed model can be transformed as Eq. (11):

(11)$$\begin{aligned} {f_r} &= \alpha \frac{{132\rho }}{{\pi \cos {\theta _i}}}\exp \left[ { - {{\left( {4\pi \frac{\sigma }{\lambda }\cos {\theta_i}} \right)}^2}} \right] + \beta \frac{\rho }{{\cos {\theta _i}\cos {\theta _r}}}{\pi ^3}{({\cos {\theta_i} + \cos \theta {}_r} )^4}{\left( {\frac{a}{\lambda }} \right)^2}{\left( {\frac{\sigma }{\lambda }} \right)^2}\\ &\quad \cdot \exp \left\{ { - {{\left( {\frac{{\pi a}}{\lambda }} \right)}^2}[{{{\sin }^2}{\theta_r} + {{\sin }^2}{\theta_i} + 2\sin {\theta_r}\sin {\theta_i}\cos ({{\phi_i} - {\phi_r}} )} ]} \right\} \end{aligned}$$

where a is the surface autocorrelation length. α and β (which equals to 1- α) are the weights of the specular reflection and scattering, respectively. Fresnel reflection coefficient ρ of the oil surface can be calculated using Eqs. (12) and (13):

(12)$${\rho _s} = {|{{R_s}} |^2} = {\left|{\frac{{{n_1}\cos {\theta_i} - {n_2}\cos {\theta_t}}}{{{n_1}\cos {\theta_i} + {n_2}\cos {\theta_t}}}} \right|^2}$$

(13)$${\rho _p} = {|{{R_p}} |^2} = {\left|{\frac{{{n_2}\cos {\theta_i} - {n_1}\cos {\theta_t}}}{{{n_2}\cos {\theta_i} + {n_1}\cos {\theta_t}}}} \right|^2}$$

where n₁ and n₂ are the reflective index of air and oil; ρ_s and ρ_p are the reflectances of orthogonal and paralleled polarization, and the Fresnel reflection coefficient is the arithmetic mean of the two phases of polarization.

As indicated in Eq. (11), the proposed model stands for the summation of reflectance induced by the specular refection and the scattering on a rough surface. It should be noted that this numerical model is valid under the following conditions: (1) σcosθ_i/a < 0.2, (2) σ/λ < 2, and (3) the incidence angle is greater than 2°.

3.2 Experiment design

Laboratory measurements were conducted to validate the proposed model. The seawater samples were collected in March 2021 at the port of Lingshui in China, on the northern Yellow Sea. As a type of commonly-used ship fuel, 0# diesel was used for the purpose of the experiment. Oil samples were accurately collected using pipette and added onto the surface of seawater samples. The oil samples were placed statically until the stable oil films are formed. It took less than 20 second to form the stable oil films.

The measurements were conducted in dark room conditions. A hemi-sphere frame was built to measure the incident and reflection angle (Fig. 3(a)). The container was wrapped with black tape to reduce interference induced by the reflection from the bottom and sides of the container. Furthermore, in order to eliminate the reflection from the ground, the container was surrounded by several blue light-absorbing sponges (Fig. 3(b)). The hyperspectral sensor applied in this study is Teledyne Lumenera Lt365R pushbroom camera, which takes a slit of 1456 pixels in each frame. The spectrum range of the camera is 392.00–1162.67 nm, with 1936 bands and the spectral resolution at about 0.4 nm.

Fig. 3. Photographs of the experiment settings: (a) hemi-sphere frame for spectra measurement; (b) container that is wrapped in black tape and surrounded by light-absorbing sponges

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3.3 Inversion of the reflectance

The inversion of reflectance involved a standard aluminum reflection panel. The reflectance spectrum of the reference panels is shown as Fig. 4. In this study, absolute reflectance (R) as a function of wavelength (λ) is obtained by multiplying the reflectance measurement that is relative to reference panel (FR_rel) by the reflectance of reference panel itself (FR_Alum) using Eq. (14).

(14)$$FR(\lambda )\, = \,F{R_{rel}}(\lambda )\,F{R_{Alum}}(\lambda )$$

According to Fig. 4, the reflectance of the reference panel is stable in the visible spectral range (400-700 nm). Considering the spectral range of the hyperspectral camera, the inversion of reflectance is also limited in the visible spectral range.

Fig. 4. Reflectance spectrum of the standard aluminum reference panel

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4. Results and discussions

4.1 Measurement and inversion of the reflectance

The experimental results under different incident zenith angles are shown in Fig. 5. The results indicate that the reflectances of diesel fuel surface are general higher than those of the clean water surface; furthermore, the reflectance spectra for sea water are relatively flat in the visible range, while those for oil samples have an absorption peak at around 600 nm.

Fig. 5. Reflectance spectra of seawater and oil surface under different wind condition.

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According to the variation of surface roughness at different windspeed, the reflectance spectra have notable differences for the seawater and diesel fuel surface under different sets of windspeed. The measurements in all wavelengths under 5 m/s are smaller than those under 3 m/s. Because as the windspeed increases, the oil surface becomes rougher. Thus, more photons are scattered into other direction and the reflectance become lower in the specular direction.

4.2 Comparisons between the model calculation and the measured data

The summation of the specular component (Eq. (8)) and the scattering component (Eq. (9)) is fitted to the experimental measurements in order to determine the additional parameter α. The fitting process is performed using an iterative algorithm of Marquardt-Levenberg method [28]. Meanwhile, we initialize σ² as the arithmetic mean of σ_x² and σ_y². The fitting statistics for our model are listed in Table 1.

Table 1. Fitting Statistic of the proposed model

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According to the fitting statistics in Table 1, the residual sum of squares (RSS) are less than 10⁻⁵, and adjusted R-squares are close to 1 under both sets of the windspeed condition. Thus, the proposed model is proved to provide accurate predication on the reflectance of rough oil surface at all the three feature wavelengths. Compared the model calculations with the measured data (Fig. 6), the proposed model provides more accurate predications at large incident angles than those at small incident angle. According to measured reflectance at different incident angle (Fig. 6), the reflectance curve is stable, and the reflectance is relatively low in small incident angle, but increases significantly with the increase of incident angle. The proposed model is able to capture this trend, and provides accurate predictions at large incident angle. When the incident angle is small, the reflectance is also small, thus the measurements could be easily affected by equipment noise and that leads to low signal-to-noise ratio in the collected data. That may explain why the model is able to provide more accurate prediction at large incident angle.

Fig. 6. Comparation of the proposed model with experimental measurements under different wind conditions and feature wavelengths: (a) 450 nm and 3 m/s; (b) 450 nm and 5 m/s; (c) 550 nm and 3 m/s; (d) 550 nm and 5 m/s; (e) 650 nm and 3 m/s; (c) 650 nm and 5 m/s.

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5. Conclusions

In this study, a numerical model for rough diesel fuel surface reflectance is developed and validated with reflectance measurements under laboratory condition. The proposed model is built on the theoretical basis of electromagnetic waves and BRDF, and featured with Helmholtz integral and Kirchhoff approximation. The reflectance of rough diesel fuel surface has been measured at the spectral range from 400 nm to 1000 nm. Reasonable agreement exists between the measured results and the model predications. It is expected that the proposed model would provide a theoretical basis for rapid detection of oil spill on rough sea surface.

In our future studies, the effects of multi-layer beam interference, polarization and ambient light will be investigated, and suitable models that take these factors into account will be developed to calculate the optical properties of oil films on seawater more comprehensively.

Funding

National Key Research and Development Program of China (2020YFE0201500); Liaoning Revitalization Talents Program (XLYC2001002); Postdoctoral Research Foundation of China (2020M670730).

Acknowledgments

The authors are grateful to Dr Chuanmin Hu from University of South Florida for the valuable suggestions on the measurements of reflection spectrum. The authors would also like to thank Dr. Zhenduo Zhang and Dr. Yu Liu from Dalian Maritime University for providing some of the oil samples.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper is available at the corresponding author on reasonable request.

References

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6. R. N. Clark, J. M. Curchin, T. M. Hoeffen, and G. A. Swayze, “Reflectance spectroscopy of organic compounds: 1. Alkanes,” J. Geophys. Res. 114(E3), E03001 (2009). [CrossRef]

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Windspeed	Fitting statistics	450nm	550nm	650nm
3 m/s	RSS^a	2.959×10⁻⁶	5.912×10⁻⁶	1.182×10⁻⁵
3 m/s	Adj. RS^a	0.954	0.960	0.875
5 m/s	RSS	2.161×10⁻⁶	2.623×10⁻⁶	4.558×10⁻⁶
5 m/s	Adj. RS	0.936	0.967	0.935

Numerically modelling the reflectance of a rough surface covered with diesel fuel based on bidirectional reflectance distribution function

Abstract

1. Introduction

2. Related studies

2.1 Modelling the optical reflectance of oil surface

2.2 Definition and applications of BRDF

2.3 Davies model of sea surface

3. Methodology

3.1 Structure of the proposed model

3.2 Experiment design

3.3 Inversion of the reflectance

4. Results and discussions

4.1 Measurement and inversion of the reflectance

4.2 Comparisons between the model calculation and the measured data

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (14)

Optics Express