International Journal of Antennas and Propagation

Volume 2019, Article ID 5363264, 15 pages

https://doi.org/10.1155/2019/5363264

## Time-Varying Ocean-Like Surface Scattering at Grazing Incidence: Numerical Analysis of Doppler Spectrum at HF/VHF/UHF Bands

Electronic Information School of Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Biyang Wen; nc.ude.uhw@newyb

Received 28 January 2019; Revised 9 June 2019; Accepted 24 June 2019; Published 15 July 2019

Academic Editor: Angelo Liseno

Copyright © 2019 Yidong Hou et al. 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

This paper numerically analyzes the characteristics of the Doppler spectrum at HF/VHF/UHF bands from 1D time-varying ocean-like surfaces at grazing incidence in vertical polarization mode. The rough surface is transformed into a local perturbation plane which has its roughness flattened at the edges. The scattering waves include coherent reflected wave and incoherent scattering waves. The surface currents exciting the incoherent scattering waves are regarded as the unknowns which can be solved from the improved surface integral equation using the method of moments (MoM). The incident plane wave allows the incident angle to reach up to 90° (grazing incidence). Then the backscattering wave in the far field can be calculated, and the Doppler spectrum is obtained by coherent Monte-Carlo simulation. Firstly, the validity of the method is verified by comparing with the mature small perturbation method at the HF band. Then the incident wave frequency is asymptotically increased from HF to UHF, and the application range of the SPM is quantitatively evaluated in the Doppler spectrum domain. Finally, the paper focuses on analyzing the characteristics of Doppler spectrum in different bands and different sea states and comparing the influence of nonlinear ocean waves on the Doppler spectrum at different frequencies.

#### 1. Introduction

In the issue of remote sensing of sea state using radar, the echo Doppler spectrum can provide more details than the radar cross-section (RCS). The Doppler spectrum from time-varying ocean-like rough surface has been studied both theoretically and numerically for the past several decades [1–4]. Benefiting from the small perturbation method (SPM) and Kirchhoff approximation (KA) theory, it has been possible to extract the sea state information from the echo Doppler spectrum at HF and X band [5–8]. However, these analytical methods will be invalid at VHF and UHF band [9, 10]. In fact, the sea echoes in the VHF and UHF bands also contain abundant sea state information. As early as 1965, [11] had used the UHF coherent Doppler radar to observe the echo Doppler spectrum of the Pacific region. In recent years, our team has used shore-based all-digital UHF radar to measure near-shore ocean echo Doppler spectrum [12], hoping to extract near-shore sea state information from the sea clutter of UHF radar. However, due to lack of theoretical basis, this work has been stagnant. Because accuracy is independent of frequency, the numerical method is a reasonable way to analyze the relationship between the Doppler spectrum and sea state at VHF and UHF band.

When using numerical methods to solve rough surface scattering problems, it is inevitable to deal with the influence of the finite rough surface edges on the scattering wave. The classical rigorous approach is to constrain incident wave with tapered beam to make it has a Gaussian footprint on the rough surface [9]. As the grazing angle becomes smaller, the tapered wave width needs to be increased exponentially to satisfy the Helmholtz equation [13]. Especially at the grazing incidence, can reach thousands of times of radio wavelength [14], which greatly increases the calculating burden. Therefore, the problem of low grazing angle (LGA) has become a stumbling block for numerically analyzing the electromagnetic scattering and Doppler spectrum from rough ocean-like surface. In order to deal with this kind of problem, many scholars have made a lot of attempts and improvements. First, a number of acceleration algorithms, including the banded matrix iterative approach (BMIA) [15], the steepest descent fast multipole method (SDFMM) [16], and the method of ordered multiple interaction (MOMI) [17], were proposed to accelerate Monta-Carlo simulations of large-scale random rough surface scattering. Paper [14] used the MOMI to simulate the Doppler spectrum from time-varying ocean-like surface at a grazing angle of and L band. Paper [18] studied 14-GHz backscattering from ocean-like surfaces at a grazing angle of with the BMIA. Another kind of approach is to process the rough surface. Paper [19] assumed that the rough surface is periodic and then used periodic boundary condition and incident plane wave to analyze rough surface scattering at LGA. However, the actual rough sea surface is aperiodic, and the effect of this assumption on the Doppler spectrum has not been evaluated. Paper [20] added the resistive loading on rough surface edges to suppress the edge current, which effectively eliminated edge effects when the LGA was greater than . Paper [21] proposed the grazing method of moments (GMoM), a novel and rigorous approach to deal with the scattering from rough surfaces under grazing illumination. GMoM represents the rough surfaces as a bounded perturbation of a plane which enlightened by a plane wave and solves the tangential components of the scattered field with improved boundary integral equation method (IEM). The effectiveness of the GMoM at LGA has been verified by comparison with both classical numerical and analytical methods. More implementation details and performance evaluations can also refer to these works [22–24].

In this paper, considering that most shore-based ocean state remote sensing radars work in the vertical polarization mode (TM case), we focus on TM case and analyze backscattering Doppler spectral characteristics from time-varying ocean-like surfaces at HF/VHF/UHF bands and grazing incidence. For simplicity and to reduce the computational burden, we restrict our study to 1D surface. Following the GMoM, ocean-like rough surface is considered as a local perturbation on an infinite plane and the roughness of the perturbation region close to zero at the edges. The surface currents producing the incoherent scattered waves, which we call scattering currents, are seen as the surface unknowns. Then the improved surface integral equation with impedance boundary condition is established to solve surface unknowns using the moment method. Using incident plane wave as excitation allows grazing incidence. The surface integral equation acts on unbounded local perturbation plane, so there is no edge effect. The scattering currents on the perturbation region also close to zero at the edges, which can be seen as windowing. The windowed scattering currents can effectively suppress the side lobes of the scattered waves. Finally, the Doppler spectrum is obtained through coherent Monte-Carlo simulations. This calculation process is very similar to the classical moment method [25]. The differences are treating the rough surface as a locally perturbed plane and replacing the total surface currents with the surface scattering currents as surface unknowns. To verify the effectiveness of these processes, the simulated Doppler spectra under different levels of sea state are compared with that calculated by the SPM at HF band. Then the simulation frequency is increased gradually from HF to UHF to quantitatively analyze the application scope of the SPM in the Doppler domain. This paper focuses on numerical analysis of the Doppler spectral characteristics of the linear and nonlinear sea surface (including Creamer model [26] and Lagrange model [27]) at different wavebands and the variation of the Doppler spectrum with sea state levels.

The rest of the paper is organized as follows. Section 2 describes the generation of linear and nonlinear time-varying ocean-like rough surface and the processing of rough surface edges. Section 3 gives the basic electromagnetic scattering formulas and the Monte-Carlo simulation of backscattering Doppler spectrum. In Section 4, the validity of the numerical Doppler spectrum at grazing incidence is verified by comparing with the SPM at HF band, and then the Doppler spectrum characteristics of linear and nonlinear ocean surface at HF/VHF/UHF bands are discussed in detail. Section 5 summarizes the conclusions.

#### 2. Linear and Nonlinear Gravity Waves

From HF to UHF band, sea waves interacting with electromagnetic waves mainly belong to gravity waves [28]. Therefore, the contribution of the capillary waves to the scattered field is not considered in this paper. In this section, we will generate one-dimensional linear and nonlinear gravity waves for the follow-up study of the paper. The nonlinear waves contain the Creamer waves [26] and the Lagrange waves [27].

##### 2.1. Linear Waves

The wave height of the linear sea surface satisfies the Gaussian distribution, which means that the linear waves can be regarded as a sum of a series of sine waves, and the phase and amplitude of each component obey the uniform distribution and Rayleigh distribution, respectively. The intensity of the sine wave with different wavenumbers is determined by the sea spectrum. Therefore, the linear gravity waves can be expressed aswhere is the wavenumber, depends on the rough surface length , is the independent complex random Gauss variable with zero mean and unit variance, is the angular frequency of sea wave with a wavenumber of determined by the dispersion relation under the deep water condition, is the gravitational acceleration, and denotes the sea spectrum. The rough surface is sampled as discrete points, and is an integer in the interval of . Then the discrete sea surface profile can be calculated using the fast Fourier transform (FFT).

In this paper, JONSWAP spectrum is employed to drive the sea waves, which is expressed bywhere is the water wavenumber, represents the wind speed at 10 meters above the sea level, is a constant representing the range of wind area, is usually 3.3, andAs an inadequately developed spectrum, JONSWAP spectrum is more realistic than the PM spectrum. When increases, JONSWAP spectrum and PM spectrum will be closer.

##### 2.2. Creamer Waves

The actual sea waves are nonlinear, and there are weak interactions between waves with different frequencies. These weak interactions can be considered as higher-order perturbation solutions to the hydraulic motion equation, which will significantly affect the characteristics of the Doppler spectrum, and the influence degree will also change with the sea state. Therefore, the nonlinearity of the sea waves cannot be neglected when studying the Doppler spectrum of the sea surface no matter which band the radar works in. The second-order nonlinear waves, generated by two wave interactions, have been described by [29, 30], which have been applied in the second-order RCS of ocean surface at HF band [31]. Although the perturbation expansion method is clear, it is not suitable for numerical implementation due to the high calculating burden especially for higher-order nonlinear waves. Another alternative method to generate nonlinear sea surfaces is the Hamiltonian formalism based on the weak wave-turbulence theory developed by [26], which has been widely used to study the electromagnetic scattering of 1D or 2D time-varying sea surfaces [3, 4].

To begin with, the Creamer method expresses the nonlinear term of the sea surface as a Hilbert transform of the linear sea surface. In the 1D case, the Creamer nonlinear term is denoted aswhere the complex amplitude is as defined in (1). This transform can also be implemented using FFT to reduce the computation cost to . In the frequency domain, the complex amplitude of Creamer nonlinear waves can be expressed asSince (5) cannot be calculated using FFT directly, it needs approximate processing to improve computational efficiency. Expanding to Taylor series, the* m*th expansion of (5) can be obtained aswhere denotes Fourier transform. Then* m*th-order Creamer waves can be calculated using inverse Fourier transform, which is It can be proved that the first-order Creamer waves just are linear waves.

##### 2.3. Lagrange Waves

Both perturbation method and the Creamer method describe the vertical skewness of sea waves in the Euler coordinate system. There is horizontal skewness in the actual nonlinear waves, which directly affects the slope distribution of the sea surface and induces more remarkable influence on backscattering signals than vertical skewness [32, 33]. References [32, 33] have attempted to modify the perturbation model to add horizontal skewness on the nonlinear waves and discussed its effect on microwave scattering and emission. Recently [27, 34] proposed a more concise method derived in the Lagrange coordinate system to generate nonlinear waves with horizontal skewness also called Lagrange waves. A disturbance is added in the horizontal direction so that an additional horizontal offset is produced in the linear waves, which iswhere and describe the vertical and horizontal movements of individual particles with time , which are two correlated Gauss processes. The vertical process of is given by (1), and horizontal process of is completely determined by the vertical process, given bywhere is the response function which determines the nonlinear characters of the Lagrange waves, in particular crest-trough and front-back asymmetry. In Lindgrens derivation [27], consists of two parts:where represents the water depth, is the sign function, and is obtained from the relation between the horizontal acceleration of water particles and vertical displacement [27]. Following the study of [27, 34], we set in this paper. The first part of determines the vertical skewness and results in waves with more peaked crests and shallower troughs (crest-trough asymmetry) compared to the linear waves, and the second part of determines the horizontal skewness and gives waves front-back asymmetry. When the water depth is assumed to be infinite, the response function can be simplified to .

##### 2.4. Wave Height and Slope Distribution

Figure 1 gives the comparison of linear and nonlinear surfaces at the wind speed of 10 m/s. The Creamer waves are accurate to the third order. In order to highlight the front-back asymmetry of Lagrange waves, the generated sea surfaces contain only waves spreading forward (). We can see that both Cramer and Lagrange waves have more flattened troughs and sharper crests, which belongs to vertical skewness. In addition, Lagrange waves also have front-back asymmetry compared to Creamer waves. The crests tilt in the direction of waves spreading.