The Mono/Bistatic SAR Imaging Simulation of Sea Surface with Breaking Waves Based on a Refined Facet Scattering Field Model

In order to analyze the scattering characteristics of sea surface under high sea state, a complete scattering model of sea surface considering breaking wave is established in this study based on the refined facet scattering field model (RFSFM) and the scattering theory of breaking wave. On the basis of this model, the influence of breaking waves on the mono/bistatic SAR imaging of sea surface at HH and VV polarization is studied. The results show that with the increase in wind speed, the coverage of breaking wave increases obviously and the consideration of breaking wave has a good correction for the scattering coefficient at HH polarization under grazing incidence; however, for VV polarization, the effect of breaking wave is very small.


Introduction
Radar plays an increasingly important role in the detection and identification of targets in marine environment. When the electromagnetic wave emitted by radar irradiates on the complex marine environment, the received scattering echo not only contains the shape, position, motion state, and other information of the target on sea surface but also inevitably contains the information of the sea surface. erefore, the establishment of a more realistic electromagnetic model to study the electromagnetic scattering characteristics of complex marine environment is of great significance to fully understand the marine environment [1][2][3][4].
As a high resolution microwave detection device, SAR is famous for its ability to obtain target image information in complex environment. By fully understanding the complex scattering mechanism of sea surface, the SAR images can be obtained to monitor the complex marine environment. Based on this, many scholars have done decades of research on the imaging mechanism of marine SAR [5][6][7][8]. Alpers [9] analyzed SAR images of ocean waves based on the velocity bunching (VB) model. Lyzenga [10] proposed a model with time dependence. Hager [11] proposed a distributed surface (DS) model, in which the radar echoes of short-wave are regarded as uncorrelated spatially. Nunziata [12] developed a SAR sea surface wave simulator based on the velocity bunching theory. Migliaccio [13] experimentally analyzed the SAR ocean speckle intensity K-distribution model versus sea surface wind field. Rizaev [14] presented a universal simulation framework for SAR imagery of the sea surface based on the linear theory of sea surface modeling, Michell thin-ship theory for Kelvin wake modeling, and ocean SAR imaging theory. Xu [15] deeply studied the simulation of SAR echo signal in complex ocean environment by using the four-path model and high frequency calculation method. Jin et al. [16] studied the bistatic scattering and transmission of the sea surface using the forward-backward method with the spectral accelerate algorithm (FBM/SAA) method. Zhang et al. [17,18] studied the bistatic SAR imagery of sea surface using the extended nonlinear chirp scaling algorithm and bistatic SAR image of composite ship-ocean scene based on the four-path model. rough the efforts of many scholars, the exploration, exploitation, and utilization of the complex marine environment have been greatly promoted.
In general, the sea surface will show complex features at high sea state including breaking waves. When the radar works at grazing incidence, the results of Bragg scattering mechanism are quite different from the actual measurement results, which cannot explain the phenomenon of sea spike and high polarization ratio caused by breaking waves [19][20][21][22]. erefore, it is of great significance to establish an electromagnetic model to study the scattering echo of sea surface with breaking waves at high sea state. In this paper, the nonlinear Creamer model is used to simulate the geometry of sea surface, and the refined facet scattering field model (RFSFM) together with the scattering theory of breaking waves is used to study the electromagnetic scattering of sea surface. Finally, the mono/bistatic SAR imaging of sea surface is simulated and discussed when the breaking waves are contained.

Position Distribution of Breaking Wave
In order to obtain the position distribution of breaking wave on the sea surface reasonably, in this paper, the geometric criterion proposed by Longuet-Higgins is used to judge whether the wave is a breaking wave; that is, when the slope at a certain position of the sea surface is greater than 0.586, it is regarded as a breaking wave [23]. It should be pointed out that in the simulation, the sea surface is discretized by many small facets with a certain size, and the size of facet will have an influence on the slope distribution of sea surface. erefore, it is necessary to select an appropriate discrete interval to simulate the sea surface to ensure the correctness of the coverage rate of the breaking waves under a certain wind speed. Monahan's empirical model [24] based on the measured data is used as a reference to verify whether the coverage rate of breaking wave obtained by simulation is reasonable, and the relationship between the coverage of breaking waves and the wind speed u 10 (m/s) at the height of 10 meters above the sea surface can be expressed as Figures 1(a) and 1(b) show the distribution of breaking points on sea surface when the wind speed is 10 m/s and 15 m/s. e 2D sea surface is generated using the nonlinear Creamer sea surface model, the sampling number is M � N � 512, and the discrete interval Δx � Δy � 0.85. From Figures 1(a) and 1(b), it could be obviously found that when the wind speed is 10 m/s, the distribution of breaking points on the sea surface is only a few, and when the wind speed is 15 m/s, the breaking points on the sea surface have gradually increased. And with the increase in wind speed, there will be a more obvious increase for the number of breaking points. Figure 2 gives the comparison between the simulated coverage of the breaking waves and the measured model at different wind speeds. From Figure 2, it could be found that with the increase in wind speed, the coverage of breaking wave increases obviously, and there is an excellent agreement between the simulation results and those obtained by the Monahan formula. So far, relatively reliable and stable breaking wave coverage at different wind speeds on the sea surface has been obtained. It can be seen that the simulation results are reasonable and effective.

EM Scattering of Nonbreaking Waves from Sea Surface.
e refined facet scattering field model (RFSFM) proposed in this paper is used to calculate the scattering coefficient of nonbreaking waves on the sea surface. In the model, the scattering region of 2D sea surface is divided into two parts: specular region and nonspecular region. e facet-based asymptotical model (FBAM) [25] and Kirchhoff approximation method (KA) are used to calculate the scattering of facets located in the nonspecular region and specular region. erefore, the scattering field of a facet can be expressed as follows: where the indexes of p and q denote the polarization of incident and scattering waves in the global coordinate system, respectively. e expression of E KA pq is shown as follows: where k i and k s are, respectively, the unit vectors along the incidence and scattering direction; k is the wave number of electromagnetic waves; η is the impedance of the free space; n is the normal vector; R is the distance between the centre of facet and the observation point; E and H are the total electric field and magnetic field on the boundary surface; and E 0 is the amplitude of incident wave. e polarization factor S pq and integral term I KA can be expressed as where ε is the relative dielectric constant of sea water; F pq is the polarization factor; and I(·) is an integral of the surface profile within a facet. e detailed expressions of F pq and I(·) can be found in [25], which are omitted here due to space limitations. en, the total field from a whole sea surface could be obtained by the summation of the scattering fields from all the facets: where M and N are the number of discrete points on the 2D sea surface, and the scattering model described in equation (2) is called the refined facet scattering field model (RFSFM). erefore, the radar scattering coefficient σ n−wb of sea surface without considering breaking waves can be obtained by

EM Scattering of Breaking Waves from Sea Surface.
Wave breaking is a complex and random process. In order to calculate the scattering coefficient of the breaking wave on the sea surface, in this paper, the model proposed by Kudryavtsev et al. [26] in 2003 is used to improve RFSFM, and then the influence of breaking waves on SAR imaging is studied. e scattering coefficient of breaking waves σ bw can be expressed as where θ is the local incident angle of the facets; s 2 wb is the mean-square slope of roughness enhancement at breaking waves; and ε wb is the ratio of the thickness to the length of the breaking unit.
According to the EM scattering model of nonbreaking waves and breaking waves on the sea surface, the scattering coefficient of whole sea surface σ sea can be obtained by where q is the coverage rate of breaking waves. Figure 3 shows the backscattering coefficients of sea surface with and without breaking waves. Figure 3(a) is with the wind speed of 7 m/s and Figure 3(b) is with 13 m/s. e incident frequency is 13.9 GHz. It can be seen from Figure 3 that the scattering coefficient at HH polarization is much smaller than the measured data [27] without   considering the breaking waves. Taking the breaking waves into consideration, the scattering coefficient of HH polarization at large incident angle is greatly improved, which is more consistent with the measured data. As for VV polarization, the scattering contribution of breaking waves is relatively small.

SAR Image Simulation and Analysis at High Sea State.
On the basis of the above model, the influence of breaking waves on mono/bistatic SAR imaging of sea surface is analyzed in this section under the conditions of different polarizations and different incidence angles. In general, SAR echo is related to the scattering coefficient of sea surface, which changes with time in the process of radar operation. erefore, the random and time-varying characteristics of the sea surface make the simulation of SAR image have a certain complexity. For this reason, many scholars have done a lot of research work and put forward some approximate theories, and the commonly used model is the velocity bunching (VB) model proposed by Alpers [9]. In this paper, the velocity bunching (VB) model is used to simulate SAR image of sea surface with breaking waves. And, the intensity of SAR image at each position can be expressed as International Journal of Antennas and Propagation where σ(x ′ , y ′ ) represents the RCS after considering tilt modulation and velocity bunching modulation; T vb is the velocity bunching modulation function; f r (·) is the range resolution function; V is the speed of the radar platform; B is the gain of the radar antenna, and because its value does not affect the relative intensity of SAR image, the influence of B is ignored in this paper; u r is the track speed along the observation direction; ρ aN ′ (x ′ , y ′ ) is the actual resolution after considering the target acceleration and limited coherent time; ρ aN is the theoretical azimuthal resolution of a stationary target; N l is the radar sight; λ is the radar wavelength; T is the integration time; τ s is the scene coherence time; and a r is the orbital acceleration along the observed direction.
If there are no special instructions, the 2D sea surface is generated using the nonlinear Creamer sea surface model, the sea surface area is 500m × 500m, and the relative dielectric constant of the sea water is calculated by the Klein model at 20°C and 35% salinity. Figures 4 and 5, respectively, give the monostatic SAR image of sea surface at HH polarization and VV polarization without and with considering breaking waves. e incident frequency is 10 GHZ, the incident angle is 70°, the wind speed is 10 m/s, the total cumulative time is 0.25 s, and R/V � 60. As shown in Figures 4 and 5, one can see that the intensity of SAR image of sea surface with considering breaking waves at HH polarization is greatly enhanced. And SAR is more sensitive to the breaking waves at HH polarization. However, in the case of VV polarization, the variation of image intensity of monostatic SAR is not obvious, which is because that the influence of breaking waves on the scattering of sea surface at VV polarization is much smaller than that at HH polarization. In addition, the position distribution of breaking waves in SAR image is not the same as its geometric position, which is caused by the motion of breaking waves. Moreover, different breaking waves have different motion states, so their position changes are also different. Figures 6 and 7, respectively, give the bistatic SAR image of sea surface at HH polarization and VV polarization without and with considering breaking waves. e incident frequency is 10 GHZ, the incident angle is 70°, the scattering angle is 60°, the wind speed is 10 m/s, the total cumulative time is 0. 25     International Journal of Antennas and Propagation monostatic SAR images in Figures 4 and 5. Namely, when the breaking waves are considered, the intensity of bistatic SAR images at HH polarization is significantly greater. However, the change of image intensity is not obvious at VV polarization. Figure 8 gives the monostatic SAR image of sea surface with considering breaking waves at large incident angle. e incident angle is 80°, and the wind speed is 10 m/s. From  Figures 8(a) and 8(b), it can be found that with the increase in incident angle, the image intensity ratio of HH to VV polarization nearly reaches 1. As the incident angle continues to increase, the ratio will exceed 1. e maximum intensity of HH polarization is greater than that of VV polarization, which shows the phenomenon of high polarization ratio and sea peak; these similar phenomena are also shown in many real SAR images. erefore, the model in this paper can be used to analyze the scattering of sea surface environment under high sea state.

Conclusion
In this paper, the improved scattering model of sea surface under high sea state was developed, which is the foundation for the investigation of the mono/bistatic SAR imaging of sea surface. With the help of RFSFM as well as the improved breaking wave model, the coverage of breaking waves on sea surface under different wind speeds is analyzed and the influence of breaking waves on the mono/bistatic SAR imaging of sea surface at HH polarization and VV polarization is analyzed in detail. e simulation results show that with the increase in wind speed, the coverage of breaking waves increases obviously, which is consistent with the situation of real sea surface. e mono/bistatic SAR is more sensitive to the breaking waves at HH polarization, and the intensity is greatly enhanced. In addition, the intensity ratio of HH to VV polarization is greater than or equal to 1 when the incident angle increases. However, at VV polarization, the graphic intensity of mono/bistatic SAR of sea surface did not increase greatly. erefore, the improved model in this paper can modify the scattering of sea surface under high sea state and has certain application value.

Data Availability
e simulation data plotted in figures used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.