International Journal of Antennas and Propagation

Volume 2016, Article ID 2474708, 14 pages

http://dx.doi.org/10.1155/2016/2474708

## Polarimetric Scattering from Two-Dimensional Dielectric Rough Sea Surface with a Ship-Induced Kelvin Wake

School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

Received 6 August 2015; Revised 17 December 2015; Accepted 20 December 2015

Academic Editor: Lorenzo Crocco

Copyright © 2016 Pengju Yang and Lixin Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on the polarimetric scattering model of second-order small-slope approximation (SSA-II) with tapered wave incidence for reducing the edge effect caused by limited surface size, monostatic and bistatic polarimetric scattering signatures of two-dimensional dielectric rough sea surface with a ship-induced Kelvin wake is investigated in detail by comparison with those of sea surface without ship wake. The emphasis of this paper is on an investigation of depolarized scattering and enhanced backscattering of sea surface with a ship wake that changes the sea surface geometric structure especially for low wind conditions. Numerical simulations show that in the plane of incidence rough sea surface scattering is dominated by copolarized scattering rather than cross-polarized scattering and that under low wind conditions a larger ship speed gives rise to stronger enhanced backscattering and enhanced depolarized scattering. For both monostatic and bistatic configuration, simulation results indicate that electromagnetic scattering signatures in the presence of a ship wake dramatically differ from those without ship wake, which may serve as a basis for the detection of ships in marine environment.

#### 1. Introduction

Fully polarimetric radars with four channels (HH, HV, VH, and VV) have advantages over the conventional single-polarization radar or dual-polarization radar when measuring ocean wave characteristics, since a fully polarimetric radar measures the entire scattering matrix, whereas a single-polarization radar or dual-polarization radar can only measure one or two elements of the scattering matrix. Ship-induced wake is of great value for the detection and classification of ships in marine environment, due mainly to the more distinct radar image signatures of ship-generated wake than ship itself. Much research effort has been devoted mainly to the extraction of specific information associated with ship from SAR images [1, 2] and to the radar imaging simulation of sea surface with ship-induced wakes [3–6]. The presence of ship-induced wake increases the roughness of sea surfaces under low wind conditions especially for a higher ship speed, and a larger sea surface roughness gives rise to a stronger multiple scattering. Fung and Eom investigated the multiple scattering and depolarization by Kirchhoff approximation and pointed out that the depolarized backscattering originates from multiple scattering [7]. Hence, it is of great value to investigate the fully polarimetric scattering including copolarized and depolarized scattering from sea surface in the presence of ship-induced wake.

Electromagnetic scattering from rough surface [8, 9] and rough surface with ship target on it [10–14] have been investigated extensively. However, fully polarimetric scattering from sea surface in the presence of ship-induced wake is rarely considered in the existing literatures, although copolarized (HH, VV) scattering from perfect electric conducting (PEC) sea surfaces with ship-induced wakes has been investigated in [15]. This is due mainly to the complexity of geometric modeling of sea surface with ship-generated wake as well as the corresponding fully polarimetric scattering modeling, which is three-dimensional (3D) electrically large problem and is beyond the capability of numerical methods such as method of moments (MoM) [16] and finite-difference time-domain (FDTD) [17]. Also, significant computational burden increases, since electromagnetic scattering from sea surface in the presence of ship-induced wake involves the stochastic properties of sea surfaces and we thus have to resort to Monte Carlo simulation. Analytical approximate models can deal with 3D electrically large problem but is limited to their validity domains [18]. Moreover, most of analytical approximate models cannot correctly predict the depolarization of wave scattering from rough surface. Among analytical approximate models, the Kirchhoff approximation [19–21] also known as the physical optics and the tangent plane approximation cannot correctly show distinct polarization dependence. The two-scale model [22, 23] also known as composite surface model underestimates the cross-polarized components due to the neglect of second-order Bragg scattering. The second-order small perturbation method (SPM) can predict the depolarization in the plane of incidence, but the validity domain of SPM is restricted to small roughness [24]. The first order small-slope approximation (SSA-I) to some extent can predict the depolarized scattering outside the plane of incidence but cannot predict the depolarization of wave scattering from rough surface in the plane of incidence [25, 26].

Voronovich and Zavorotny [27] pointed out that the depolarized scattering from rough sea surface arises from two effects. The first effect is a result of mutual transformation of the two linear polarization states caused by facets’ tilts and the second one is due to the second-order Bragg scattering. The second-order small-slope approximation (SSA-II) takes into account the two effects and thus can predict the electromagnetic wave depolarization from rough ocean surface. Accordingly, SSA-II is used in this paper to predict the monostatic and bistatic polarimetric scattering from sea surface in the presence of ship-induced Kelvin wake. The major concern in the present study is to investigate the depolarized scattering and enhanced backscattering from rough sea surface in the presence of ship-induced Kelvin wake by comparison with single rough sea surface without ship’s wake. The remainder of this paper is organized as follows. Section 2 briefly presents the modeling of sea surfaces in the presence of ship-induced Kelvin wake. The polarimetric scattering model of SSA-II with tapered wave incidence is briefly presented in Section 3. The numerical results of monostatic and bistatic polarimetric scattering from sea surfaces with and without ship’s Kelvin wake are discussed and analyzed in Section 4. Section 5 concludes this paper.

#### 2. Modeling of Sea Surface with Ship-Induced Kelvin Wake

The simulation of rough sea surface elevation is crucial for the modeling of electromagnetic scattering from rough sea surface. The linear sea surface instead of nonlinear sea surface is considered in the present study, since the nonlinearity of ocean waves mainly influences the Doppler spectrum of time-varying rough sea surface and the present study involves only the average scattering coefficient of sea surface. The linear superposition method, fractal theory, and spectral method are widely used in generating linear rough sea surface.

In the present study, the spectral method under spatially homogeneous and time-stationary hypothesis is used to generate rough sea surface by taking into account its high efficiency, since the procedure of generating rough sea surface can be realized by fast Fourier transformation. All of the components of the linear sea surface in Fourier domain are completely independent of random phases. This implies Gaussian statistics for the sea surface elevations and their derivatives. The spatial Fourier amplitude at any time can be expressed as follows:where is spatial wave vector. is the sea surface roughness spectrum. is a complex Gaussian random series with zero mean value and unit variance. The sampling interval along direction is which is determined by the Nyquist sampling criterion with being the length of sea surface along direction. Similarly, with being the length of sea surface along direction. According to the gravity-capillary dispersion relation, the angular frequency is related to the spatial wavenumber by with being the gravity acceleration constant.

Then, the sea surface elevation at spatial potion for time can be expressed by

Equation (2) can be efficiently accomplished by inverse fast Fourier transformation. To ensure that is real, is required to satisfy the conjugate symmetry about the origin as follows:

The sea surface roughness spectrum proposed by Elfouhaily et al. is used in the present study for generating sea surface, which consists of gravity waves spectrum and capillary waves spectrum and is expressed as [28]where and denote the long-wave and short-wave curvature spectrum, respectively. The detailed expressions of and can be found in [28]. represents the angular spreading function and is chosen as follows:where is the wind direction with respect to the radar line of sight andwhere , rad/m, with being inverse wave age, m/s, and is the friction velocity.

Based on the Michell thin ship theory combined with the classic theory of a ship’s wave resistance with a ship being regarded as a rigid body moving in inviscid incompressible fluid, Zilman et al. derived an explicit asymptotic expression for ship-generated Kelvin wake in the far zone [6]. Here, we briefly present the main formulations associated with a ship’s Kelvin wake.

To approximately simulate a ship’s wake, a parabolic ship is used which depends on length , beam , and draft :

The ship-generated elevation of the free surface is related to the fluid velocity potential by the following expression:

Zilman et al. derived the asymptotic expression for fluid velocity potential by using a parabolic ship depicted by (7) as follows:where is the Froude number. It should be noted that (8) combined with (9) is in the form inverse Fourier integral and can be efficiently calculated by using fast Fourier transformation.

The total wave elevation of the free surface is assumed as a superposition of wind-generated sea surface elevation and ship-induced free surface elevation as follows:

Figures 1(a) and 1(b) present ship-induced Kelvin wake elevation of free surface with ship speed m/s and wind-generated sea surface elevation of free wave with wind speed m/s, respectively. Under a fixed wind speed m/s, the simulated sea surfaces with ship-induced wake are presented in Figures 1(c) and 1(d) for ship speeds m/s and m/s, respectively. From Figure 1, we can observe that the sea surface geometric structure is changed due to the presence of ship-induced Kelvin wake. More precisely, the sea surface roughness becomes larger due to the existence of a ship’s Kelvin wake, especially at low wind conditions. The electromagnetic scattering signatures from sea surface will be thus affected due to the presence of a ship’s Kelvin wake, which will be examined in detail in what follows.