Scattering Characteristics of Targets above a Rough Surface in SAR Images

e modeling, simulation, and analysis of target characteristics are essential to a synthetic aperture radar (SAR) image-based autotarget recognition (ATR) system.e coupling effect between targets and rough surface is also important to the electromagnetic scattering and remote sensing. In this work, the simulations to SAR images of targets above a �nite rough surface have been investigated. e effect of rough surface on the target characteristics, or the coupling between the rough surface and targets, is analyzed in details by observing changes of locations and intensities of scattering centers in the SAR images. e SAR images are obtained by taking two-dimensional inverse fast Fourier transforms (FFTs) of the scattered �elds, which are computed by the combined high-frequency method of shooting and bouncing ray (SBR) and truncated-wedge incremental-length diffraction coefficients (TW-ILDCs). Simulated results of SAR images for complicated targets above a rough surface are given under the 0.25 × 0.25m resolution at the X band, in which the coupling effect between targets and rough surface has been studied in details.


Introduction
e classi�cation to the region of interests (ROIs) or the scattering features in synthetic aperture radar (SAR) images is the �nal and most important stage in a SAR-image-based autotarget recognition (ATR) system.e simulation of SAR images has signi�cant importance to the image understanding and target recognition.Compared to the inverse SAR (ISAR) images of targets in the free space such as airbornes, the simulations of SAR images to various vehicles on the ground or ships on the sea are much more complicated and difficult.In such simulations, the effects of ground or sea surface to the scattering from targets need to be considered, which is important to the real ATR system.
In the past decades, the approaches on how to take into account the coupling between the ground or sea surface and targets have been extensively studied.Johnson calculated the scattering of dielectric objects in a half space using the four-path method [1].Guan et al. computed the scattered �elds of a perfectly electrical conducting (PEC) target above a �nite PEC rough surface using the method of moment (MoM) and half-space Green's function [2].Xpatch could be used to calculate the scattering by objects-either PEC or dielectric-above a dielectric rough surface using the high-frequency (HF) methods and generate the SAR images rapidly [3][4][5].However, the parameters of the ground or sea surface, including the dielectric constant, and electric conductivity could not be enumerated.On the SAR image simulations, Franceschetti et al. developed a method to produce SAR raw signals, including the stripe and spotlight mode based on the Kirchhoff approximation (KA) method [6].Depending on this method, the ground or sea surface has been modeled as a rough surface divided by quadrilateral meshes, and the scattering from every mesh conforms to the Gaussian distribution.Xu and Jin generated a set of SAR images of wider scene with different vegetation and buildings using a mapping and projection algorithm (MPA) [7].
Under the point-target hypothesis, a real target consists of many scattering points or light spots in its SAR image.However, the light spots are not always appearing inside the SAR image, while they would change with the incident frequency and attitude angle of the target.Another obvious feature in the SAR image is that a shadow exists on the back to the radar sight.e shape of the radar shadow is similar to the shadow when illuminated by light but is not equal to that.e radar shadow is fuzzier than lighting shadow because of the coherent imaging mechanism and the resolution ratio.In the ATR system, the frequently used features include the location and intensity of the scattering centers, the contour of the target, and the shadow.However, the above features of a target alone will be different to those of the target above the ground or sea surface, while the latter will be used in the real ATR systems.
Hence the coupling between the surface and target must be considered when studying the target features in SAR images.In this paper, the frequency-domain scattered �elds of targets above a rough surface are computed using the shooting and bouncing ray (SBR) technique and the truncated-wedge incremental-length diffraction coefficient (TW-ILDC) method.e SAR images of targets are then calculated by taking a two-dimensional (2D) IFFT on the scattered �elds, in which the coupling effects have been investigated and interpreted in details.When taking IFFT, the resampling and window function methods are used to preprocess the raw frequency-domain data.
e paper is organized as follows.e SBR technique and TW-ILDC method are introduced in Section 2.1, and the simulations of SAR images are investigated in Section 2.2.Numerical examples are presented in Section 3, which include the targets in free space, the rough surface itself, and the coupling between targets and rough surface.Finally, a conclusion is made in Section 4 to illustrate the importance of the simulations in the real ATR system.

Computation of Scattered Fields. Aer developed by
Ling et al. [8] to calculate the scattering of cavity, the SBR technique had been used to solve the scattering problems for large electrical-size targets successfully.Xpatch is a wellknown soware combining SBR and other effective methods to calculate the radar cross sections (RCSs), SAR images, and ISAR images of targets.e main idea of SBR is to use a dense grid of shooting rays as the incident �eld, then calculate the multiple re�ections of every shooting ray by the geometrical optics (GO) method when the ray reacts to the target, and �nally to compute the scattered �elds using the physical optics (PO) method when the ray escapes from the target.e most time consuming part in SBR is not the electromagnetic (EM) computing but the ray tracing, which could be accelerated by means of parallel methods or GPU techniques.e disadvantage of SBR is that it does not include the diffraction and creeping waves, which would in�uence the accuracy of computation.e equivalent PO current on the surface of a PEC target is written as in which the coefficient  accounts for the shadowing effects, and   is the outward unit normal vector to the illuminated object surface at ′.We could calculate the scattering electrical �eld by where ( − ′) is the scaler Green's function in free space To improve the accuracy of the SBR technique, the TW-ILDC method has been considered to include the diffractions from �nite sized wedge faces [9,10].e TW-ILDC method has been con�rmed to be useful in improving the accuracy, especially for the complicated targets.e detailed derivation of this method and the numerical results have been presented in [11,12].e total scattered �eld is the superposition of the SBR �eld and the diffraction �eld as follows: In general, the frequency-domain scattered �eld could be computed by the combined SBR and TW-ILDC methods [11].

e SAR-Image Generation.
It has been proved that the SAR image could be generated from the frequencydomain scattered �elds using the 2D-IFFT algorithm [13].e incident magnetic �eld is obtained by   () = ()  ×   from the speci�c incident electric �eld.en the scattered �eld is written as ( We suppose that the   component of the scattering �eld is received, and de�ne a shape function (′).en the   component of the scattered �eld is written as from which the relationship between the scattered �eld and the Fourier transform of the shape function is obtained as follows: in which  () is the Fourier transform of the shape function at is to say that the shape of the target could be obtained from the scattered �eld by using IFFT [13].
In this paper, the 2D SAR image will be considered, which is the projection of the shape function on the imaging plane, the    plane.Hence the relationship between the 2D SAR image and the shape function is From the above equations, it is clear that we could obtain the SAR images from the scattered �elds simulated by the SBR and TW-ILDC methods.First, a set of scattered �eld data are simulated.en a similar procedure is conducted to the PFA step in the spotlight SAR imaging, which takes an interpolation on the data before performing IFFT.e window function method is applied before IFFT to reduce the Gibbs' effect [14].To obtain a high-resolution SAR image, a set of wideband and wide-azimuth data are needed.e resolution in the range and azimuth,   and   , is to be determined as where  is the light velocity,  is the bandwidth of the electromagnetic wave,   is the central frequency, and Δ is the rotation angle of SAR.e simulated data are distributed over a sector region in the wave number domain (WND) on the      plane.Although the distribution region is similar to a rectangle, it is still different (the azimuth is large).Hence a sector-torectangle interpolation, which is called as resampling in PFA, is needed before IFFT.Once the resolutions are con�rmed, the resampling numbers could be computed as follows: where  max is the maximum length of the simulation scene.
On the other hand, to reduce the Gibbs' effect, we use a 2D Taylor window function on the resampled data in the  domain [14].

Numerical Examples
e main purpose of this paper is to illustrate the effect of a rough surface to SAR images of targets located above the rough surface.e Gaussian distribution rough surfaces has been widely used in the scattering research, which is chosen as ground model.Because the SBR method is used to calculate scatter �eld of models, the rough surface is also set as PEC objects.e rough surface is described as the Gaussian spectrum by in which   and   are the correlation lengths in -and directions, respectively, and ℎ is the rms height of the rough surface.e generation progress has been shown in [15].In the SBR method, the simulation time is related to the size and mesh density of the targets.Hence an in�nite rough surface is impossible for simulations and we have to choose a �nite-sized rough surface.e model should be static PEC or dielectric coated electrically large ground targets and the size of targets should be larger than the minimum resolution cell in the image.It also could be meshed before calculation.e incident parameters are de�ned in Figure 1(a) and the size of rough surface is set in a minimum length as shown in Figure 1(b) because it could contain the shadow area which could be an important feature in an ATR system.
Here are the explanation of the multiple scattering routes on the rough surface.In Figure 1(b), we could see that there are several scattering routes to contribute to the back scattering, which are (1) the direct re�ection by the rough surface; (2) the direct re�ection by the target; (3) multiple scattering from the target to rough surface; (4) multiple scattering from the rough surface to target.
ere are also other multiple scattering that could be ignored.Each route could contain numbers of re�ections during SBR method [8].In this paper, the number of re�ections could be 2 or 3 according to different models and could satisfy the calculation accuracy [11,12].
In the �rst example, the effect of a rough surface to targets is investigated through the simulation of a SLICY-like target above a rough surface which is shown in Figure 2. SLICY is a vehicle-size target which includes canonical shapes and cylindrical surfaces.e forward direction is looking into the trihedral corner, which we call the front of the target.e maximum dimensions of SLICY-like target shown in 2 are 815  in width, 1258  in length, and 647  in height.In the following simulations, all the rough surfaces are discretized by �at triangles whose maximum edge length is 3  ( is the wavelength in free space).
e target is located above a Gaussian rough surface with   =   = 15  and ℎ = 2 .e horizontal dimensions of the rough surface are set as 350   350 .e plane waves are incident from the direction of  = 75 ∘ and  = 0 ∘ , 15 ∘ , 30 ∘ , 45 ∘ , 60 ∘ , 75 ∘ , 90 ∘ ), respectively, the central frequency is 10 GHz, and the bandwidth is 1.5 GHz.Here, the incident direction implies the central direction of SAR simulation because   directions in the range of Δ are required.From (10) we can calculate that the range resolution is 0.1 m.We also suppose that the azimuth resolution is equal to the range resolution, so that Δ is 429546 ∘ .
Figure 3 shows the SAR image results of SLICY-like target on the Gaussian rough surface for the HH polarizations where -axis represents range direction and -axis represents cross-range direction.e horizontal streak in Figures 3(a), 3(b), 3(g), and 3(h) is mainly caused by the Gibbs' effect during IFFT, especially in Figures 3(a) and 3(g).An obvious feature of Gibbs' effect in SAR image is sidelobes next to the strong scattering points.Results of VV have similar characteristics, actually.To illustrate the effect of scattering from the rough surface, the corresponding SAR images of  the target in free space are also simulated.Because the main strong scattering effects are brought by the �rst three re�ections, the multiple re�ections more than 4 are ignored.Limited by space, only the results of    ∘ , 3 ∘ , 6 ∘ , 9 ∘ ) are demonstrated here, which are clear enough to show the rough-surface effect.
e strong scattering points in SAR images are generated mainly by the dihedral angles and trihedral angles on the target.When SLICY is placed above a rough surface, there is a local "dihedral angles" which supply multire�ections constructed by the bottom of target and rough surface so that there are obvious differences between the results of free space and above the rough surface, as illustrated in Figures 3(a), 3(b), 3(g), and 3(h).Although the "dihedral angles" are different to those constructed by the target and a PEC plane, the construction still have similar multire�ection route during ray tracing in SBR method especially enhancing the backscattering.Meanwhile, in Figures 3(c), 3(d), 3(e), and 3(f), although there are multire�ection constructions at the bottom of the target, the second re�ection will not return back because the direction of the re�ection is different to the backscattering.Comparing to the multire�ection part, the backscattering of the rough surface is so small that it could not be displayed clearly in the images.
To compare the differences of the results in free space and above the rough surface, we calculate the correlation coefficient of the two images in all of the 7 directions.e correlation coefficient is de�ned as corrcoef   ) −  )  ) where  and  are the two image matrices.e corrcoef is close to 1 when  and  are similar to each other, and if    then corrcoef  1. e curve of the correlation coefficient is shown in Figure 4.
From Figure 4 we can see that the correlation coefficients are close to 1 except two special directions at    ∘ and 9 ∘ .Results in Figures 3 and 4 illustrate that the rough surface always impact the SAR image results by changing the local consistence of re�ectors.�owever, the other re�ectors located on the plane of the SLICY-like target are not in�uenced by the rough surface so that the correlation coefficients at directions of   1 ∘ , 3 ∘ , 4 ∘ , 6 ∘ , 7 ∘ ),   7 ∘ are close to 1.
In an SAR-based ATR system, the strong scattering points are the most important features that would be used during the matching procedure.Next we present simulation results of a complicated target, a model of tank, to illustrate the SAR images in free space and above a rough surface and the changes of amplitudes and locations of scattering points when the target is above different rough surfaces.ere are also tank results that could be obtained in [4,5].e size of tank model is 29  in length, 17  in width, and 73  in height.e three-dimensional (3D) model of the tank above a Gaussian rough surface is shown in Figure 5. Firstly, we consider the tank model above different Gaussian rough surfaces.Generally, the coupling is weak for the parts of rough surfaces which are far from the target because small quantity of rays could reach the target.It is important to con�rm the size of the Gaussian rough surface.If the size of the rough surface is small, the accuracy of the scattering effect could not be guaranteed; if the size is very large, however, huge amount of simulation time would be required.Here, we consider two sizes of rough surfaces, which are 1000   1000  and 1250   1250 , respectively.e correlation length and height of the two rough surfaces are both   =   = 15  and ℎ = 5 .e direction of the tank gun barrel is set as 0 ∘ azimuth direction, and the three incident directions are set as  = 5 ∘ and  = 0 ∘ , 45 ∘ , 60 ∘ ]. e central frequency of the incident HH-polarization plane wave is 9.5 GHz, and the bandwidth is 0.6 GHz.e azimuth range is .61 ∘ .According to (10), the resolutions of the range and cross-range are   =   = 0.25 m.
Figure 6 shows the SAR image results of the tank model above different Gaussian rough surfaces.From top to bottom, each row means different imaging azimuth directions of the target, which are 0 ∘ , 45 ∘ and 60 ∘ , respectively.e le column of Figure 6 is the result with smaller rough surface, and the right column is the result with larger rough surface.e display ranges in the SAR images are the same, but are different from those when simulating since they need different samples to satisfy the Nyquist's sample law.e images are zoomed in to �t the size of tank model to be displayed more clearly.e grounds are generated randomly so that the noise is different.
From Figure 6 we can see that the brightest pixels are nearly the same when the tank target is above different-size rough surfaces.In the azimuth of 0 ∘ , the brightest pixels represent the tank wheels, and the next bright pixels located in the right of the brightest ones indicate the oil drums and hatch, which are nearly the same in the two SAR images, as shown in Figures 6(a) and 6(b).e weak scattering points such as the front cover and gun barrel are different in two images because the scattering of these parts is so weak that could be easily submerged by the scattering of rough surfaces.In the azimuth of 45 ∘ , the brightest pixels represent the wheels and hatch part of the tank, and the next bright pixels are located in the two points of front cover on the right of the hatch.ese features are nearly the same in the two SAR images demonstrated in Figures 6(c) and 6(d), but a tiny difference exists in the gun-barrel scattering point.It is brighter in Figure 6(c).
For the case of 60 ∘ azimuth angle, the SAR image results in such two situations have signi�cant difference.is is because the energy is not concentrated to a few strongscattering points but distributed in many weaker pixels.Hence the scattering by the rough surface could in�uent the amplitudes and locations of these pixels signi�cantly.ere are several obvious differences in the two SAR images, as illustrated in Figures 6(e) and 6(f).First of all, the point located at the back cover appears in Figure 6(f) but disappears in Figure 6(e).Secondly, the two strong scattering points generated by the hatch and toolbox are clearer in Figure 6(f).Regardless of such differences, the distributions of strong scattering points in such two images are basically similar to satisfy the requirement of an ATR system.
Figure 7 shows another SAR image results of the tank model above different rough surfaces, in which the incident angle is ( = 5 ∘ ,  = 45 ∘ ).e sizes of rough surfaces are 800   800 ,     , 106   106 , 1200   1200 , and 1   1 , respectively.Other parameters are the same as those in Figure 6.
From Figure 7, we can see that the �ve �gures are nearly the same, which means that the coupling between the tank model and rough surface is only concentrated around the tank model.Hence it would be accurate enough if the size of the rough surface is set to 1000   1000 .
In the above examples, the tank target and the rough surface are regarded as a combined system in the numerical simulations.To investigate the coupling effect of rough surface, several "superposed" SAR images are generated by superposing the free-space target images on the roughsurface images.e correlation coefficient is also calculated to illustrate the similarity of the two kinds of images.We present several simulations to illustrate the difference between the tank target in free space and that above a Gaussian rough surface.e size of the rough surface is 1   1 , and other parameters are the same as the earlier example.Figure 8 shows the results of "combined" and "superposed" SAR images of the tank target.e le column is the "superposed" SAR images, while the right column is the "combined" SAR images, respectively.From top to bottom, the azimuth directions are  ∘ , 4 ∘ , and 6 ∘ , respectively.
From Figure 8 we can see that the obvious difference between the two kinds of SAR images is whether a shadow of the tank exists or not.e "combined" method is based on the ray-tracing technology, hence the occluded part of the rough surface by the tank could not be illuminated by rays.Besides, there are other differences in which we have more interest.
In the azimuth of 30 ∘ shown in Figure 8(b), an extra strong scattering point exists at the right-hand of gun barrel, which disappeared in Figure 8(a).In the azimuth of  ∘ , the two SAR images have a difference on the scattering points of wheel locations, which implies that the rough surface and the tank con�rm a local re�ection structure which changes the route of re�ected rays.In the azimuth of 60 ∘ , the strong scattering points are basically the same, but there is an extra point at (21,16) in Figure 8(f).is point is caused by the multiple re�ections from the tank to the rough surface, which do not exist in Figure 8(e).

Conclusions
In this paper, the scattering characteristics of targets above a Gaussian rough surface in SAR images are investigated via

F 1 :
e incident parameters de�nition and the minimum size of a rough surface.

F 2 :
e geometrical model of a SLICY-like target above a Gaussian rough surface.

F 3 :F 4 :
e 2D HH-polarization SAR images of the SLICY-like target in free space and above a Gaussian rough surface, in which the incident direction is (   ∘ and    ∘ , 3 ∘ , 6 ∘ , 9 ∘ ]), the resolution is �.� m, and the ma�imum re�ection number is 3. (a) e SAR image of target in free space at the azimuth of    ∘ .(b) e SAR image of target above a Gaussian rough surface at the azimuth of    ∘ .(c) e SAR image of target in free space at the azimuth of   3 ∘ .(d) e SAR image of target above a Gaussian rough surface at the azimuth of   3 ∘ .(e) e SAR image of target in free space at the azimuth of   6 ∘ .(f) e SAR image of target above a Gaussian rough surface at the azimuth of   6 ∘ .(g) e SAR image of target in free space at the azimuth of   9 ∘ .(h) e SAR image of target above a Gaussian rough surface at the azimuth of   9 ∘ .International Journal of Antennas and Propagation e correlation coefficients of the SAR images of the SLICY-like target in free space and above a rough surface at 7 azimuth directions.F 5: e 3D geometrical model of a tank above a Gaussian rough surface.

F 6 :
e 2D HH-polarization SAR images of the tank target above Gaussian rough surfaces with different sizes, in which the incident angle is (   ∘ ,    ∘ , 4 ∘ , 6 ∘ ]), the resolution is �.2� m, and the maximum re�ection number is 2. e gun barrel direction is set as    ∘ .(a) e SAR image of the tank above the smaller rough surface at azimuth of  ∘ .(b) e SAR image of the tank above the larger rough surface at azimuth of  ∘ .(c) e SAR image of the tank above the smaller rough surface at azimuth of 4 ∘ .(d) e SAR image of the tank above the larger rough surface at azimuth of 4 ∘ .(e) e SAR image of the tank above the smaller rough surface at azimuth of 6 ∘ .(f) e SAR image of the tank above the larger rough surface at azimuth of 6 ∘ .

F 7 :
e 2D HH-polarization SAR images of the tank target above Gaussian rough surfaces with different sizes, in which the incident angle is (   ∘ ,    ∘ ), other parameters are the same as in Figure 5. (a) e SAR image of the tank above rough surface of 800   800 .(b) e SAR image of the tank above rough surface of 933   933 .(c) e SAR image of the tank above rough surface of 106   106 .(d) e SAR image of the tank above rough surface of 1200   1200 .(e) e SAR image of the tank above rough surface of 1333   1333 .