Investigation of the electromagnetic (EM) scattering of timevarying overturning wave crests is a worthwhile endeavor. Overturning wave crest is one of the reasons of sea spike generation, which increases the probability of false radar alarms and reduces the performance of multitarget detection in the environment. A threedimensional (3D) timevarying overturning wave crest model is presented in this paper; this 3D model is an improvement of the traditional twodimensional (2D) timevarying overturning wave crest model. The integral equation method (IEM) was employed to investigate backward scattering radar cross sections (RCS) at various incident angles of the 3D overturning wave crest model. The super phenomenon, where the intensity of horizontal polarization scattering is greater than that of vertical polarization scattering, is an important feature of sea spikes. Simulation results demonstrate that super phenomena may occur in some time samples as variations in the overturning wave crest.
A considerable amount of research has recently focused on sea spikes, which are a matter of great importance. Sea spikes have been found to cause false target detections and are typically characterized by horizontal polarization (HH) signals that exceed vertical polarization (VV) signals by as much as 10 dB or more [
LONGTANK waves [
Numerical techniques have been widely used in recent research on sea surface scattering. However, these techniques involve long computational time and large memory requirements; thus, numerical techniques have become research bottlenecks, especially when considering highfrequency 3D scattering problems. Although several integral equationbased techniques [
The remainder of this paper is organized as follows. Section
The improved timevarying overturning wave crest was modeled according to [
The construction of the 2D timevarying overturning wave crest model will be described in detail at first. As is known, the Beaufort wind scale is always used to describe the sea condition, which includes the wind speed and the wave height. The sea condition for the different Beaufort wind scales and the relationship between the Beaufort wind scale and wind speed have been presented in [
Furthermore, the relationship between the wave height and the wind speed can be obtained by the Gaussian function fitting method and can be approximately written as [
For the 2D timevarying overturning wave crest model, the length and height of the wave crest are varying with the time stepping. Therefore, the time factor was included to control the profile of the overturning wave crest model; the 2D timevarying overturning wave crest model can be expressed as
Computational parameters.

Overturning  After overturning  







2  0.7  2  0.5 


0.25  0  0.25 


0 



4  0  4  40 



1  1 

16  4  16  4 





The 3D model was obtained by stretching the 2D overturning wave crest model in the
2D and 3D overturning wave crest models.
2D model
3D model
For the EM scattering of the 3D timevarying overturning wave crest, the multiple scattering plays a major role in the scattering results because of the steep part of the model. However, the traditional Kirchhoff method is difficult to deal with this problem, because the multiple scattering is not considered. Therefore, IEM is utilized to study the EM scattering of improved timevarying overturning wave crest model.
Considered an incident plane wave, it can be expressed as
In the IEM, the governing equations for the tangential surface fields on a dielectric surface can be written as the sum of the standard Kirchhoff surface field and a complementary surface field, which can be written as [
According to [
For the IEM, a complementary surface field was added to the Kirchhoff surface field to improve the resulting accuracy. However, the Kirchhoff surface field is widely used to determine the EM scattering of rough surfaces and is valid when the surface roughness is large and the surface slope is small, while if the surface slope is large, the related multiple scattering will be significant and the Kirchhoff surface field becomes invalid. Moreover, the complementary surface field part represents the scattering from the other facets. Therefore, the IEM obviously has wider applications than the Kirchhoff method.
The improved timevarying overturning wave crest model should be meshed into a large number of triangles to determine EM scattering by the IEM; the EM scattering process is divided into three steps.
When the direction of incident wave is given, if a triangle can be lighted by the incident wave, it should meet two necessary conditions: At first, the triangular facet should not be sheltered by itself—
Figure
Shelter determination.
In addition, it should be noted that the determination of the lighted triangles is timeconsuming. Therefore, octtree [
The validity of the IEM results for EM scattering of the 3D overturning wave crest model was compared with that of the finite element method (FEM), which was performed on FEKO. FEM is a numerical technique, and FEKO is an EM simulation software program that is widely used for EM computation. The 3D overturning wave crest model is shown in Figure
Comparison between the IEM, KA, and the FEMFEKO.
Figure
Profiles of the 16 overturning wave crest models in the
Profiles of 16 overturning wave crest models in
Figures
Backscattering RCS of 16 3D overturning wave crest models (upwind).
Backscattering RCS of 16 3D overturning wave crest models (downwind).
Figure
Figure
Figure
The difference value of the backscattering RCS between HH polarization and VV polarization for the 16 3D overturning wave crest models.
As presented in Figure
In this paper, an improved timevarying overturning wave crest model was described in detail. In this model, the influence of wind speed on the size and height of overturning wave was considered. The element of time was also discussed to control the overturning wave crest model profile. The corresponding 3D overturning wave crest model was then obtained by stretching the 2D overturning wave crest model in the
Based on the geometrical features of the 3D overturning wave crest model, we employed the IEM to investigate the backscattering RCS of 16 3D overturning wave crest models and reveal the process of overturning wave crest generation. Simulation results show that the sea spike phenomenon occurs in some time samples, and this result is attributed to the instantaneous profile of the overturning wave crest model as well as the incident angle. The overturning wave crest is one of the reasons of the sea spikes, and the sea spikes are more likely to occur for the upwind incidence.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation for Distinguished Young Scholars of China (Grant no. 61225002) and the Aeronautical Science Fund and Aviation Key Laboratory of Science and Technology on Avionics Integrated Sensor System Simulation (Grant no. 20132081015).