AN ELECTROMAGNETIC MODEL TO DESCRIBE BACKSCATTERING FROM AN OIL/WATER MIXTURE Buono, Andrea; Nunziata, Ferdinando; Migliaccio, Maurizio Dipartimento di Ingegneria, Università degli Studi di Napoli “Parthenope”, Centro Direzionale, isola C4 – 80143, Napoli ABSTRACT In this study, an electromagnetic model to describe the scattering from an oil/water mixture is proposed. To take in account the volumetric scattering due to oil inclusions in the sea water, an equivalent medium characterizing the mixture has been developed. The electromagnetic behaviour of the oil/water mixture has been modeled using the MaxwellGarnett (M-G) approximation formula. Then, to predict the mixture surface scattering, the Improved Integral Equation Method (IIEM), has been used. Preliminary numerical experiments, undertaken with particular reference to the widely diffused cases of natural oil seeps, have been accomplished for different oil volume fractions in the mixture and for frequencies ranging from 1 to 10 GHz. 1. INTRODUCTION Among various marine scenarios, one of the most complex from an electromagnetic modeling perspective is the oil/water mixture. When dealing with oil seeps or particular oil spills, the interaction between oil and sea water cannot be reduced to a conventional surface scattering model, such as the Bragg one. A volumetric scattering model is to be developed to describe the contributions coming from a water column, where different kinds of oil inclusions are present. An electromagnetic model in such situation is not straightforward, also due to weathering processes that can alter the oil/water mixture (i. e. evaporation and emulsification). This study arises from the fundamental importance to better understand the scattering mechanisms that lie at the basis of such a problem, and to assist Synthetic Aperture Radar (SAR)-based techniques for oil spills monitoring. In fact, this study aims at investigating the complex scattering which characterizes an oil/water mixture, with particular reference to the natural oil seeps. They are usually coastal geologic features which frequently occur off the southern coast of California and in the Gulf of Mexico, characterized by the natural low-rate flow of liquid or gaseous hydrocarbons (see Fig. 1). The topic is very interesting not only from a scientific point of view, but also from an economical and environmental perspective. In fact, one may think that in the Gulf of Mexico there are more than 600 offshore natural oil seeps releasing into the marine ecosystem up to 5 million barrels of oil per year. An electromagnetic model of such complex scenario has to take in account the volumetric scattering due to presence of fresh oil inclusions continuously leaked into the water column. In this study, the electromagnetic behaviour of the oil/water mixture has been modeled using the MaxwellGarnett (M-G) approximation, which consists ofconsidering an equivalent medium composed by both oil and sea water. Hence, to describe the scattering from such mixture, the Improved Integral Equation Method (IIEM) have been employed. Preliminary numerical experiments are undertaken considering the case of an oil/water mixture arising from a natural oil seep. This test case is particularly interesting since it represents an unconventional oil slick: in fact, the continuously released fresh oil coming from the bottom of the ocean not only generates a thin film over the sea surface but it also forms a water column where oil droplets are present. Figure 1: This illustration shows the route traveled by oil leaving the sub-seafloor reservoir as it travels through the water column to the surface and ultimately sinks and falls out in a plume shape onto the seafloor where it remains in the sediment. (Illustration by Jack Cook, Woods Hole Oceanographic Institution). 2. MIXTURE CHARACTERIZATION AND BACKSCATTERING MODEL In order to model the backscattering of an oil/water mixture forming a water column in which oil droplets are included, two main assumptions are due which are consistent with the natural oil seeps case: • Oil droplets in the mixture can be considered as homogeneous and spherical-shaped; • The radius of each oil inclusion is smaller than the distance between a couple of them. This latter assumption allows neglecting multiple scattering effects and it applies when the oil volume fraction in the mixture is small. Under these assumptions, volumetric scattering effects can be treated replacing the oil/water mixture with an equivalent homogeneous medium that allows considering only a surface scattering. In an Effective Medium Approximations (EMA) framework, the Maxwell-Garnett approximation is adopted to model the macroscopic properties of the composite material formed by the oil/water mixture, describing it with a proper dielectric constant [1]. In particular, the M-G model consists of considering the oil/water mixture as a certain volume fraction of inclusions (oil droplets) included in a host medium (water), as [1]: 𝜀!"" = 𝜖! ! !!!! !! !(!!!! )!! !!!! !! !(!!!! )!! !! ! ! 𝑒 !! ! !!" ! ! !!" ! ! !!! 𝜎 !! 𝐼!" ! ! (!) !!" ! !!" ,!!" ! !!" !! 3. PRELIMINARY RESULTS In this Section, preliminary results obtained running the model at different frequencies and for different oil volume fractions are discussed. The behaviour of the oil/water medium dielectric constant for different frequencies has been analyzed, given the oil volume fraction. L-, C- and X- band frequencies are considered (1.7, 5.3 and 10 GHz, respectively), with oil volume fraction increasing from 10 to 90%. The latter is shown in Fig. 2, in which real (in blue) and imaginary (in red) parts of the 𝜀!"" defined in eq. (1) show the same trend for all the oil concentrations: they have a minimum at C-band and a maximum at L-band. Morevoer, their absolute values decrease as the oil volume fraction increases. It should be also noted that the difference between the real and imaginary part 𝜖!"" , given the frequency, became smaller when increasing 𝛿! . Hence, given the frequency, this implies a variation of the skin depth according to the oil volume fraction. Similar considerations can be drawn for results obtained for intermediate values of 𝛿! , which are not shown to save space. (1) where 𝜖! and 𝜖! are the dielectric constant of sea water and oil droplets, respectively, and 𝛿! is the oil volume fraction in the mixture. Hence, the oil/water mixture is electromagnetically described by the dielectric constant 𝜀!"" of the equivalent homogeneous medium. Furthermore, it must be pointed out that the Maxwell-Garnett model is self-consistent [1]. Once the effective dielectric constant is obtained, the IIEM is used to model the backscattering from randomly rough surfaces. This model overcome sthe traditional simplifying assumptions applied to the phase of the Green’s function in the IEM leading to a more complex model but still expressed in algebraic form [2]. In fact, traditional IEM has been developed to bridge the gap between the large scale Kirchoff Approximation and the small scale Small Perturbation Method, showed by two-scale scattering models. However, neglecting multiple scattering under the aforementioned hypothesis, IIEM predicts a Normalized Radar Cross Section (NRCS) given by [2]: 𝜎!" = incident and scattered wave polarizations, respectively, and 𝐼 (!) are the source fields. More details on IIEM can be found in [2]. , (2) where the subscripts i and s stand for incident and scattered directions, respectively, 𝑊 (!) represents the n-th power Fourier spectrum of the correlation function, p and q are the Figure 2: Mixture permittivity for L-, C- and X-band for oil volume fractions equal to 0.1 (top) and 0.9 (bottom), respectively. In Fig. 3 and 4 the real and imaginary parts of the oil/water mixture equivalent permittivity spectrum, respectively, are shown for the same previously considered values of 𝛿! . For the real part, they have the same shape varying the oil concentration, showing a reduced variation rate when increasing 𝛿! . The imaginary parts, instead, follow an exponential behaviour reaching a constant value at about 3 GHz with the same rate, independently of the oil volume fraction. In Fig. 5 the behaviour of the oil/water mixture NRCS, simulated using the IIEM, is shown in decibels (dB) for the copolarized HH and VV scattering channels. For all the consi- dered bands the NRCS shows a strongly decreasing trend with the angle of incidence. tained from numerical experiments undertaken for different kinds of oil and for different oil volume fractions in a wide frequency range, confirm the soundness of the proposed approach based on the M-G approximation and the IIEM for surface electromagnetic scattering. Figure 3: Plot of the real part of the mixture permittivity evaluated The co-polarized channels are always practically overlapped until 30°, exhibiting a departure from each other at larger incidence angles. Nonetheless, the slight departure that is in place for higher frequencies (C- and X-bands) becomes very large at L-band. However, Fig. 5 witnesses that backscattering from the oil/water mixture is larger for lower frequencies. Figure 5: Oil/water mixture NRCS predicted according to the IIEM obtained using the M-G 𝜀!"" . From left to right: L-, C- and X-band results. 5. REFERENCES [1] A. H. Sihvola, Self-Consistency Aspects of Dielectric Mixing Theories, IEEE Trans. Geo. Rem. Sens., Vol. 27, no. 4, 1989. [2] K. Fung, W. Y. Liu, K.-S. Chen, and M.-K. Tsay, An Improved IEM Model for Bistatic Scattering from Rough Surfaces, Journal of Electromagnetic Waves and Applications, Vol. 16, no. 5, pp. 689–702, 2002. Figure 4: Plot of the imaginary part of the mixture permittivity evaluated for 𝛿! equal to 0.1 (top) and 0.9 (bottom), respectively. 4. CONCLUSIONS In this study, an electromagnetic model to describe the scattering from an oil/water mixture is proposed. This model aims at providing a better understanding of the the electromagnetic behavior of an oil/water mixture. This is done with particular reference to natural oil seep cases, in which a complex scattering problem is in place due to the continuous flow of fresh oil from the bottom of the ocean forming oil droplets into the sea water column. Preliminary results, ob-

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