## Abstract

Oil slick thickness was an important parameter for estimating oil spill volume. Two-beam interference theory could be used to interpret the behavior of reflected and refracted light in oil slick. A quantitative relationship between thickness and spectral reflectance of oil slick could be established based on this theory. Some parameters have the properties of numerical oscillation and can be ignored in practical application. In addition, numerical approximation results showed that two parameters of the relationship were closely related to the spectral reflectance of background water and the thick oil slick. Therefore, a practical model for estimating oil slick thickness could be derived and proved to be consisted with theoretical relationship.

©2012 Optical Society of America

## 1. Introduction

Marine oil spill accident could produce different impacts on marine ecological environment and social economy [1, 2]. Large-scale oil spill accident severely affects the marine environment, such as Deep-water Horizon oil spill accident in the Gulf of Mexico in 2010 [3]. As soon as the oil spill entered into seawater, it begins to spread by gravity, wind and current to more area and form the oil slick with different thickness on the ocean surface [4]. Estimating oil slick thickness plays an important role in assessing oil spill volume.

Remote sensing technologies provided a variety of methods to detect oil spill, such as multi/hyper remote sensing [5, 6], thermal infrared [7], synthetic aperture radar (SAR) [1], laser fluorescence [8], and multi-angular remote sensing [9]. Oil slick with different thickness showed different visual characteristics and led to different spectral reflectance within the visible light wavelength [10, 11]. Therefore, optical remotely sensed data could be used to estimate oil slick thickness. Light interference theory has been used to describe the behavior of the incident light in oil slick layer [12, 13]. Two reflected light beams superimpose and result in two-beam interference on the focal plane of the detector, and then the relationship between oil slick thickness and remotely sensed data could be established based on light interference theory [13].

In this research, oil slick floated on a calm ocean surface can be regarded as an optically flat between air and seawater. Two – beam interference theory has been used to interpret the behavior of the refracted and reflected light when the incident sun light passed through the oil slick, and then a relationship between thickness and spectral reflectance of oil slick could be established. The values of some parameters were very small and oscillatory. Therefore, those parameters could be ignored in the practical model. Numerical approximation analysis results indicated that another two parameters were closely related to the spectral reflectance of background seawater and very thick oil slick. A practical model could be established to describe the quantitative relationship between oil slick thickness and its reflectance. Moreover, this paper discussed the detectable thickness range of oil slick and the optimal optical remote sensing band when using this model to estimate oil slick thickness.

## 2. Two-beam interference theory

Oil slick floated on the sea surface can be considered to be the middle of three layers, where the upper layer is air and the lower layer is seawater. The refractive index of the air layer is ${n}_{1}$, the oil slick is ${n}_{2}$. The thickness of oil slick is $d$. The refractive indexes of these layers are different, thus the reflection and refraction characteristics of light are different. Two-beam interference theory (showed in Fig. 1 ) has been used to describe the behavior of sun light in those layers, and the theoretical relationship between the spectral reflectance and the thickness of oil slick [12].

Assume the electric field strength of sun light is $\stackrel{\rightharpoonup}{{E}_{0}}$, and the wave number is $k=2\pi /\lambda $. When the light is incident from air to oil slick, the reflectivity of oil slick upper surface is${R}_{12}$ and the transmissivity is ${T}_{12}$. The electric field strength of incident light reflected between the air and oil slick is $\stackrel{\rightharpoonup}{{E}_{1}}={R}_{12}\stackrel{\rightharpoonup}{{E}_{0}}$, the electric field strength of the transmitted light is ${T}_{12}\stackrel{\rightharpoonup}{{E}_{0}}$. When the transmitted light reflected from the lower oil slick layer, the reflectivity of oil slick lower surface is ${R}_{23}$, the electric field strength of the transmitted light that reflected by oil slick lower surface is ${T}_{12}{R}_{12}\stackrel{\rightharpoonup}{{E}_{0}}$. When the light is incident from oil slick to air, the transmissivity of oil slick upper surface is ${T}_{12}$, then the electric field strength of the transmitted light that transmitted from oil slick to air is $\stackrel{\rightharpoonup}{{E}_{2}}$. Assuming the $\stackrel{\rightharpoonup}{{E}_{1}}$ and $\stackrel{\rightharpoonup}{{E}_{2}}$ have

different propagation path lengths, and use $k\Delta $ to indicate the phase delay. The electric field strength $\stackrel{\rightharpoonup}{{E}_{2}}$could be written as $\stackrel{\rightharpoonup}{{E}_{2}}={T}_{12}{R}_{23}{T}_{21}{e}^{ik\Delta -as}\stackrel{\rightharpoonup}{{E}_{0}}$ ($a$is the extinction coefficient of oil slick). Two electric field vectors ($\stackrel{\rightharpoonup}{{E}_{1}}$and $\stackrel{\rightharpoonup}{{E}_{2}}$) superimpose and result in two-beam interference on the focal plane of the optical remote sensor. The electric field strength of oil slick detected by sensor is $\stackrel{\rightharpoonup}{E}=\stackrel{\rightharpoonup}{{E}_{1}}+\stackrel{\rightharpoonup}{{E}_{2}}$, the spectral reflectance of oil slick ($R$) can be written as Eq. (1).

Equation (1) can be written as Eq. (3).

Equation (3) describes the theoretical relationship between spectral reflectance, oil slick thickness, angle of refraction, interface reflectance and transmittance. In this research, angle of refraction, interface reflectance and transmittance can be regarded as the fixed values. The independent variable is oil slick thickness ($d$) and the dependent variable is oil slick spectral reflectance ($R$), Eq. (3) can be rewritten as:

WhereIn Eqs. (4) and (5), ${A}_{0}$-${A}_{4}$ can be regarded as five fixed values parameters. ${A}_{0}$_{,${A}_{1}$} and ${A}_{4}$are related to interface reflectance and transmittance of all the interfaces. ${A}_{2}$ is related to the extinction coefficient of oil slick.

## 3. Example simulation

The equation of two-beam interference theory is too complex; therefore, a simple practical model is important for researchers to estimate oil slick thickness. In this study, a laboratory experiment has been designed to obtain the oil slick thickness and spectral reflectance. The data should be used to simulate the parameters of two-beam interference theory and build a practical model. The crude oil and seawater were obtained from Liaohe Oil field and Bohai Sea in China. The seawater in beaker which outer wall and bottom were wrapped with black plastic bags to prevent light transmission has been used to simulate background. Two 50 W bromotungsten lamps were used to simulate the incident light. We used a glass injector to generate artificial oil slick, the thickness of oil slick which almost fully covers the seawater of one drop oil is about 1.2 μm based on statistics. A high resolution spectroradiometer (ASD FieldSpec ProFR) was used to collected spectral reflectance of oil slicks (the spectral resolutions are 3 nm in the range from 350 nm to 1050 nm).

Figure (2) showed that the spectral reflectance of seawater and oil slick which thickness change from 1.2 μm to 54.0 μm. Previous researches indicated that the oil slick remote sensing bands were in visible wavelength, but the optimum optical bands varied for the background difference [5,6,10,11]. In case 1 water, blue and green bands are optimum optical bands for oil slick detection [5], in case 2 water, green and red bands are optimum optical bands [6].

## 4. Results and discussion

#### 4.1 Parameter analysis

On the basis of experimental data and Eqs. (4) and (5), the values of parameters (${A}_{0}$-${A}_{4}$) for wavelengths from 381 nm to 760 nm could be calculated. Equation (4) can be regarded as consists of three parts (${A}_{0}$, ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ and ${A}_{4}{e}^{-2{A}_{2}d}$). ${A}_{0}$ is a fixed value and related to reflectivity of oil slick upper surface (${R}_{12}$). ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ and ${A}_{4}{e}^{-2{A}_{2}d}$ are related to oil slick thickness ($d$). The value range of $\mathrm{cos}({A}_{3}d)$ is [-1, 1], therefore, ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ would produce numerical concussion in Eq. (4). We chose eight bands (400nm、450nm、500nm、550nm、600nm、650nm、700nm、750nm) to analyze the change trends of ${A}_{0}$, ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ and ${A}_{4}{e}^{-2{A}_{2}d}$ with the increase of oil slick thickness.

Figure 3 showed that the change trend of ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ with the increasing of oil slick thickness. The values of${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ became smaller and tend to 0. In those bands, the largest value range is [-0.001, 0.001].

Figure 4 showed that the change trends of ${A}_{0}$, ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ and ${A}_{4}{e}^{-2{A}_{2}d}$ with the increasing of oil slick thickness. The values of ${A}_{0}$ are same in one band and greater than that of ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$. The values of ${A}_{4}{e}^{-2{A}_{2}d}$ became smaller and tend to 0 with the increase of oil slick thickness. For smaller oil slick thickness, the values of ${A}_{4}{e}^{-2{A}_{2}d}$ are greater than that of ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$. When the oil slick thickness is increasing, the values of ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ and ${A}_{4}{e}^{-2{A}_{2}d}$are all becoming smaller and tend to 0. It is assumed that the fluctuating values of ${A}_{1}{e}^{-{A}_{2}d}\mathrm{cos}\left({A}_{3}d\right)$ are fixed in practical application. Therefore, the values of it could be ignored in new simple relationship. The Eqs. (4) and (5) could be written as:

WhereEquation (6) is a new simple relationship which approached to Eq. (4) and has three parameters (${a}_{0}$,${a}_{2}$,${a}_{4}$). The physical interpretation of these parameters of Eq. (7) are same to that of Eq. (5) (${A}_{0}$,${A}_{2}$,${A}_{4}$). Therefore, if the values and changing trends of these parameters are similar, the modified Eq. (6) would be consistent with the theoretical formula. We used the same experimental data (reflectance and oil slick thickness) to calculate the five parameters of Eq. (4) and three parameters of Eq. (6). Figure 5 showed that the change trends of ${a}_{0}$,${a}_{2}$ and ${a}_{4}$ are consisted with ${A}_{0}$,${A}_{2}$ and ${A}_{4}$, and the most similarly values of parameters are in the range from 500 nm to 650nm. The results indicated that the new simple relationship (Eq. (6)) was consisted with the theoretical relationship (Eq. (4)) in the visible wavelength range.

#### 4.2 Theory and approach

The oil slick thickness optical remote sensing model could be established based on Eqs. (6) and (7). The detecting capability of optical remote sensing technique for oil slick thickness determining is limited. In this paper, it is assumed that the detection range of oil slick thickness is (0, D). D is the maximum oil slick thickness which could be determined by optical remote sensing. Analyzing the oil slick thickness approached to 0 or D value was important for us to establish the mathematical model.

### 4.2.1 Oil slick thickness approach to D value

When the oil slick thickness approach to D value, the incident light could not transmit the oil slick layer, and the two - beam interference theory would not stand. Therefore, the Eq. (1) could be written as follow:

Where ${R}_{oil-\mathrm{max}}$ was the spectral reflectance of maximum thick oil slick which could be identified by optical remote sensing.

In Fig. (6) , solid line represented the values of ${a}_{0}$ which has been calculated by using the experimental data and Eq. (6), and the dotted line was the measured spectral reflectance of thick oil slick. There are good linear relationship between the values of ${a}_{0}$ and the spectral reflectance of thick oil slick. The multiple correlation coefficients of the curve fitting method are 0.9492. Figure (6) showed that the parameter (${a}_{0}$) was a fixed value and approached to the spectral reflectance of thick oil slick in theory when the oil slick thickness approached to D value.

### 4.2.2 Oil slick thickness approach to 0

When the oil slick thickness decreased and approached to 0, the two – beam interference theory still worked, and the Eq. (6) could be written as follow:

_{${a}_{2}$}was a fixed value showed in Eq. (6), when the oil slick thickness ($d$) approached to 0, we can deduce:

The Eq. (9) can be written as

We can deduce that the parameters ${a}_{0}$ and ${a}_{4}$ are fixed values in Eq. (11), therefore, the Eq. (11) can be rewritten as

When the oil slick thickness approached to 0, we assumed the spectral reflectance of oil slick approached to that of the background seawater. Equation (13) can be derived as follow:

Where ${R}_{water}$ was the spectral reflectance of background seawater.Based on the Eqs. (8) and (13), the spectral reflectance of oil slick could be simulated when the oil slick thickness approached to 0. The relative error (*RE*) between $\underset{d\to 0}{\mathrm{lim}}R$ and ${R}_{water}$ can calculated by using Eq. (14):

Figure 7 showed that in the spectral range from 380 to 579 nm, the values of (${a}_{0}+{a}_{4}$) were greater than the spectral reflectance of background seawater; in the 580 nm band, the value of (${a}_{0}+{a}_{4}$) was equal to the spectral reflectance of background seawater; in the spectral range from 581 to 760 nm, the values of (${a}_{0}+{a}_{4}$) were less than the spectral reflectance of background seawater. Relative error analysis showed that the relative error values were less than 0.05 when the wavelength of visible light is longer than about 525 nm.

The differences between the simulated values of (${a}_{0}+{a}_{4}$) and the spectral reflectance of background seawater were produced by background seawater. When the oil slick thickness decreased and approached to 0, the influence of spectral reflectance change of background seawater has not been considered in this model. When the wavelength of visible light is shorter than about 580 nm, the seawater is characterized by low absorption, low reflection and high penetrability for incident visible light. In the blue light band, the Rayleigh scattering of seawater which is a kind of forward scattering is strong. Therefore, the influence of Rayleigh scattering of seawater on the reflectance of oil slick is nonlinear. The incident light which could be reflected by oil slick will reduce gradually when the oil slick thickness decreased and approached to 0. Consequently, the simulated values of (${a}_{0}+{a}_{4}$) would greater than the measured values in those bands. A reflectance peak in 580 nm is formed by suspended particle matter (SPM) in water. In yellow light band, the Mie scattering of SPM in water could increase the backscattering light. The influence of Mie scattering on the reflectance of oil slick is linear, therefore, the best fitting band simulated to the reflectance of oil slick was located in the reflectance peak of SPM in water. In red light and near-infrared bands, the water is characterized by high absorption, low reflection and low penetrability. The absorption characteristics of water will produce the influence of reflected light on the reflectance of oil slick. Peaks appeared in 665 nm and 750 nm showed the water’s absorption. Therefore, the relative error curve is the fluctuant in the range from 581 nm to 760 nm.

### 4.2.3 Model and applicability

On the basis of two-beam interference theory and approach, an optical remote sensing model could be established to describe the relationship between the oil slick thickness and their spectral reflectance. Based on Eqs. (6), (8) and (13), we have

Where ${R}_{oil-\mathrm{max}}$ was the spectral reflectance of the thickest oil slick; ${R}_{water}$ was the spectral reflectance of background water; $R$was the spectral reflectance of oil slick, $d$ was the oil slick thickness; a was the extinction coefficient of oil.We use the oil slick thickness as the independent variable in Eq. (15) to calculate the function. Parameter (${a}_{2}$) is calculated using Eq. (6). The wavelength band in 580 nm is used to verify and evaluate the precision of the research model.

Figure 8 showed that the simulated reflectance value and the measured value are in good agreement for the 580 nm wavelength band. As oil slick thickness increases, the simulated values tend to the same value. Therefore, we can determine the D value in our experiment is about 35 μm. The multiple correlation coefficient between measured and simulated reflectance is 0.9458 when the oil slick thickness less than D value. This research model efficiently relates the spectral response of the offshore oil slick to its thickness.

## 5. Conclusions

(1) Two-beam interference theory can be used to preferably interpret the behavior of incident light in oil slick floated upon water surface. Based on the theoretical relationship, parameter analysis and approach, a practical model could be established to describe the quantitative relationship between oil slick thickness and spectral reflectance. Extinction coefficient (a) of oil, spectral reflectance of thick oil slick and background seawater were the important parameters in this optical remote sensing model.

(2) The thickness range of oil slick estimated by this model is limited. If the oil slick thickness was greater than the certain thickness, the model could not be used to estimate the thickness. The different oil have different extinction coefficient, therefore, the maximum oil slick thickness of different oil which can be detected by optical remote sensing technology may be different.

(3) The spectral reflectance of background seawater was important for this model. The optimal optical remote sensing band has been determined by the maximal reflected peak of the background spectrum. High reflectance of background seawater helps to increase the detection ability and estimation accuracy of the oil slick thickness optical remote sensing model.

(4) The influence of scattering and absorption of background seawater under oil slick on the spectral reflectance of oil slick should be considered in the future research. If the change of reflectance of water under oil slick could be estimated, then we can improve the applicability and the estimation precision of the model.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 41001196), the Open Research Fund of Key Laboratory of Marine Spill Oil Identification and Damage Assessment Technology, SOA (Grant No. 201212), and the Open Research Fund of Key Laboratory of Digital Earth, Center for Earth Observation and Digital Earth, Chinese Academy of Sciences (Grant No. 2010LDE007).

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