AN ELECTROMAGNETIC MODEL TO DESCRIBE

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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