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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.

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 [

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 [

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 [

The rest of the paper is organized as follows. Section

From HF to UHF band, sea waves interacting with electromagnetic waves mainly belong to gravity 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 as

In this paper, JONSWAP spectrum is employed to drive the sea waves, which is expressed by

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 [

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 as

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 [

Figure

Comparison of linear and nonlinear wave profiles.

In terms of wave profiles, the difference between linear and nonlinear waves is very small. This weak difference is mainly reflected in the wave height and slope distribution and can significantly affect the radar echo Doppler spectrum. Figure

Comparison of linear and nonlinear wave height and slope distribution.

Wave height distribution

Slope distribution

Limited by the edge effect of finite rough surface, the MoM cannot directly deal with the rough surface scattering problem under near-grazing incidence. Although adopting the tapered wave as incident wave can solve the large incident angle problem well, the computation load increases exponentially with the increase of the incident angle [

In this paper, we mainly study the characteristics of backscattering Doppler spectrum from one-dimensional time-varying ocean-like rough surface. Considering that most sea state monitoring radar is deployed on the coast and operates in vertical polarization mode, we focus on the vertical polarization incident wave at the grazing incidence. When using numerical methods to analyze rough surface scattering problems, the rough surface needs to be truncated to finite area. On the one hand, the discontinuous edge current of the truncated rough surface will affect the distribution of the scattered field. On the other hand, the scattering field can be approximated as the Fourier transform of the rough surface current, and the truncated surface current will generate side lobe at the undesired scattering angles, which is similar to the spectrum leakage.

In order to deal with the edge effect, the rough surface is seen as a local perturbation on an infinite plane. We window the truncated rough surface with a Tukey-Hanning window function to construct the locally perturbed surface. As shown in Figure

Geometry of the scattering problem and the locally perturbed surface.

For vertically polarized wave, the magnetic field has only component in the

In general, after the surface equivalent electric current

Now we consider the directly truncated rough surface as a perturbation on an infinite plane and use the Tukey-Hanning window to restrain the rough surface so that its roughness fades to zero at the edge, as described in (

As shown in Figure

Generate time-varying linear waves, Creamer waves, and Lagrange waves

Use the Tukey-Hanning window to restrain the rough surfaces to construct local perturbation planes

Employ the moment method to solve the improved magnetic field integral equation to calculate incoherent backscattered waves

The time-ordered sequence of backscattered waves is collected to produce the Doppler spectrum using Fourier transform

Repeat the above steps to obtain multiple Doppler spectra for smoothing

Sketch map of the Doppler spectrum simulation.

In this section, we will discuss in detail the characteristics of the Doppler spectrum backscattering from one-dimensional linear and nonlinear time-varying ocean-like surfaces at grazing incidence. The simulation frequencies range from HF to UHF bands. Table

Simulation conditions.

Parameters | Value |
---|---|

Simulation frequency | HF to UHF bands |

Incident angle | |

Scattering angle | |

| |

Rough surface width | |

Scattering area width | |

Asymptotical area width | |

Plane area width | |

Maximum Doppler frequency | |

Simulation frame | 1024 |

Monte Carlo simulation times | 32 |

Figure

Surface current distribution over wave profile. (a) The incoherent surface current

The SPM is widely recognized for analyzing sea surface scattering problems in HF band. The first- and second-order backscattering RSC of the rough sea surface have been derived by Barrick using the SPM [

Figure

Comparison of Doppler spectrum between the numerical simulation and the SPM in different sea states. The frequency is 10 MHz and the sea surfaces are linear. The Doppler frequency is normalized by the Bragg frequency

Wind speed 5 m/s

Wind speed 10 m/s

Wind speed 15 m/s

Wind speed 20 m/s

The consistency of numerical simulation and the SPM in HF band shows that the method in this paper is effective for grazing incidence scattering. Next we will demonstrate the effective frequency range of the SPM by numerical simulation. At the same time, in order to verify the effectiveness of the proposed method in VHF and UHF bands, the classical integral equation method (IEM) is also employed for comparison, which calculates the surface equivalent currents by directly solving the EM integral equation using the moment method, and uses tapered incident wave, expressed by (

Comparison of Doppler spectrum of linear waves among the proposed method, the classical integral equation method, and the SPM in different frequency bands. The wind speed is 10 m/s, the Doppler frequency is normalized by the Bragg frequency

10 MHz

30 MHz

60 MHz

100 MHz

200 MHz

400 MHz

The mean absolute error of the Doppler spectral intensity over operating frequency between the SPM and the numerical method. The sea waves are linear and the wind speed is 10 m/s.

From Figure

Numerical simulated Doppler spectrum of linear waves over operating frequency. The horizontal axis is the Doppler angular frequency normalized by the Bragg frequency, and the vertical axis represents different simulation operating frequencies from 5 MHz to 400 MHz.

Wind speed 3 m/s

Wind speed 7 m/s

The actual sea surface is nonlinear, and the nonlinear components generated by the interaction of gravity waves will introduce hydrodynamic contribution in Doppler spectrum. Figure

Doppler spectrum comparison between linear and nonlinear waves in HF/VHF/UHF bands. The wind speed is 10 m/s, and the Doppler frequency is normalized by the Bragg frequency

10 MHz

60 MHz

200 MHz

600 MHz

In the issue of remote sensing of sea state using radar, one is more concerned about the characteristics of echo Doppler spectrum in different sea states. Figure

Doppler spectrum of nonlinear waves in different sea states and different operating frequencies. The horizontal axis represents the Doppler frequency normalized by the Bragg frequency

Creamer waves at 10 MHz

Creamer waves at 60 MHz

Creamer waves at 600 MHz

Lagrange waves at 10 MHz

Lagrange waves at 60 MHz

Lagrange waves at 600 MHz

A more intuitive comparison of the Doppler spectra in the HF and UHF bands at different wind speeds is shown in Figure

Doppler spectrum of Creamer waves in different sea states and different operating frequencies. The Doppler frequency is normalized by the Bragg frequency

10 MHz

600 MHz

In this paper, we use the improved integral equation method to study the characteristics of Doppler spectra backscattered from linear and nonlinear sea waves in HF/VHF/UHF bands at grazing incidence. In order to eliminate the edge effect and suppress the side lobes of scattered waves, the time-varying rough sea surface is transformed into a local perturbation plane by multiplying a Tukey-Hanning window function, which flattened asymptotically at the edges. Unlike the classical integral equation method, we divide the surface currents into a known coherent current and unknown incoherent currents. The coherent current excites the reflected wave, while the incoherent currents excite the scattered waves. The incoherent currents are considered as the surface unknowns and calculated by solving the surface integral equation established on the unbounded local perturbation plane using the moment method. The incident plane wave ensures the accuracy of the solution of the integral equation at grazing incidence. The calculation result shows that the intensity of the incoherent currents decreases with the decrease of roughness of the local perturbation plane and is equal to zero at the edge. The comparisons with the SPM on the one hand verify the effectiveness of the method in this paper at grazing incidence, and on the other hand, it also measures the applicable frequency range of the SPM in the Doppler spectrum domain. In the HF band, the Doppler spectrum of the numerical simulation is completely consistent with that derived by the SPM whether in the low sea state or the high sea state. The difference between the Doppler spectra derived from SPM and simulated by numerical method increases rapidly with the increase of operating frequency, which is on the one hand because the slope and wave height of the sea surface gradually fail to meet the conditions of the SPM, and on the other hand, the third- and higher-order perturbation solution to the scattered waves cannot be ignored.

The numerically simulated Doppler spectrum is robust in VHF and UHF bands, which facilitates further analysis of the characteristics of the Doppler spectrum. We compared the Doppler spectra of linear and nonlinear sea waves at different operating frequencies and different sea states. The nonlinear waves include Creamer waves with only crest-trough asymmetry and Lagrange waves with both crest-trough and front-back asymmetry. In general, the high-order peak intensity of the nonlinear waves is about 10 dB higher than that of the linear waves, and the intensity of the Bragg peak is not affected by the nonlinearity of the sea surface. In the Doppler frequency range of

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no competing interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant 61371063 and the National Key Research and Development Program of China under Grant 2017YFC0405700.