Electromagnetic Scattering Analysis for Two-Dimensional Gaussian Rough Surfaces with Texture Characteristics Using Small-Slope Approximation Method

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1 Prgress In Electrmagnetics Research B, Vl. 63, 89 31, 15 Electrmagnetic Scattering Analysis fr Tw-Dimensinal Gaussian Rugh Surfaces with Texture Characteristics Using Small-Slpe Apprximatin Methd Rng-Qing Sun *, Jing Xie, and Yang-Wei Zhang Abstract This paper is aimed at analyzing the electrmagnetic EM scattering frm the twdimensinal -D Gaussian rugh surfaces characterized by textures. Visual appearances f the stripe texture can be generated thrugh the angle rtating in Furier transfrm when the rati f the crrelatin lengths in tw directins is large enugh. The scattering field is derived in Cartesian crdinate system thrugh the small-slpe apprximatin SSA methd with plane incident wave. The nrmalized c-plarized radar crss sectin NRCS frm -D Gaussian rugh surface characterized by textures is calculated. In particular, several numerical results shw the influences f incident angle, texture angle, crrelatin length, and rt-mean-square height n the scattering frm the textured rugh surface. Finally, the validity f the SSA methd is verified by cmparisns f theretical value and measured data. 1. INTRODUCTION Research n the electrmagnetic EM scattering frm randm rugh surfaces is widely emplyed in the fields f glbe remte sensing, sea research, surface detectin and radar imaging [1 3]. In actual applicatin, the data f EM scattering are mainly frm practical measurement and theretical deductin. Fr practical measurement, it is necessary t put in a lt f manpwer and material resurces in an aggressive envirnment. Hwever, the thery mdel [4, 5], which is frm theretical deductin and maes cmparisn with experimental data, has sme advantages, such as simple realizatin methd, wea envirnment impact, which are extremely valuable in applicatin. At present, there are tw majr inds f scattering theries f rugh surfaces, i.e., numerical methd and apprximate methd. Numerical methd has very high calculatin precisin but is mre cmplicated and difficult t be realized. In this case, it is f the necessity t appeal t the apprximate methd. In the analysis f apprximate thery, the classical methds include Kirchhff apprximatin KA, small perturbatin methd SPM and tw-scale methd TSM which cmbines KA with SPM [6 9]. Cncerning KA, tangential plane apprximatin frms its basis, in which the curvature radii f rugh surface are much larger than incident wave length s that the hypthesis f EM wave incident n the infinite plane tangential t a pint n the rugh surface is hld. In ther wrds, the KA methd is apprpriate fr large scale f rugh surface but ut f place fr lw grazing incidence, whereas SPM is suitable fr a slightly rugh surface [1]. Due t the respective applicatin ranges f bth KA and SPM based n statistical mdels f rugh surfaces, there exists very great limitatin. Further, TSM has extended the applied area f scattering f rugh surfaces, but its defect lies in that the cncept f the cutff wave number, whse determinatin is lac f scientific basis, is intrduced t distinguish between large-scale and small-scale rugh surfaces when being calculated. Fr this reasn, it is necessary t find a thery that can accurately slve EM scattering f rugh surfaces withut cnsidering their structures. In this case, there appeared t be Received 1 August 15, Accepted 3 Octber 15, Scheduled 9 Octber 15 * Crrespnding authr: Rng-Qing Sun sunrngqing@16.cm. The authrs are with the Aerspace Research Institute f Materials and Prcessing Technlgy, Beijing 176, China.

2 9 Sun, Xie, and Zhang relevant numerical methds, including extended bundary cnditin methd EBCM, Mnte Carl methd MCM, finite difference time dmain methd FDTD and apprximate methd SSA [11 14]. Amng them, SSA is an effective calculatin methd and applied t any wavelengths and rugh surfaces, and it is a mre precise apprximate methd in which the expressins f different rders deduced are btained by retaining the term number f the series expansins and can be degenerated int the results f KA and SPM in certain cnditin. In recent years, a series f research results have been btained in the statistical mdels f EM scattering frm rugh surfaces [15]. Jhnsn and Warnic [16], in detail, cmpared gemetric ptics GO with physical ptics PO in the area f EM scattering thery f the rugh surface with expnential crrelatin features and cnfirmed the effective cnditins f PO thrugh the cmparisn f the calculatin results with accurate slutins. Hu et al. [17] cmpared and discussed the effects f RMS height and crrelatin length f rugh surface n the effective cnditins f several apprximatin methds. Additinally, sme schlars perfrmed research n deterministic mdel f EM scattering frm randm rugh surface. Li and Xu [18] studied scattering and Dppler spectral analysis fr -D linear and nnlinear sea surfaces by the SSA methd. Texture characteristics can reflect image prperties and visual rugh degree f rugh surface in the remte sensing images [19, ]. They are btained by analyzing the stripe pattern distributed in the regular r irregular ways. S the texture characteristics are the fundatin t describe and recgnize images in bth thery and applicatin. The distinct texture characteristics are displayed in sandban, crdillera and sea wave. Currently, the research n texture prperties is mainly frm the literature f radar image, but study f the EM scattering fr textures is much less. Praash et al. [1] nly studied the variatin f the specular scattering as sil textures at X band. S there is the necessity that the relatin between EM scattering and Gaussian rugh surface with texture prperties is established. The remainder f this paper is rganized as fllws. In Sectin, frmulatin fr -D Gaussian rugh surface, whse texture angles are adjustable, is briefly presented thrugh the angle rtatin in Furier transfrm, and discrete analysis n the rugh surface is perfrmed. In Sectin 3, hw the EM scattering calculatins are carried ut by using the first-rder SSA SSA-I and secnd-rder SSA SSA-II methd is addressed. As the SSA methd is an apprximate thery based n the expansin f the surface slpe, the secnd-rder SSA will give a mre accurate predictin f the scattering prblems, especially fr crss-plarizatin cases. In Sectin 4, the impacts f the scattering angles, incident angles, texture angles, RMS heights and crrelatin length n the single and average bistatic NRCS and the bacscattering cefficient are discussed.. REALIZATION OF -D GAUSSIAN ROUGH SURFACES WITH TEXTURE The linear filter methd, which cmbines finite impulse respnse filtering FIR with fast furier transfrm FFT, is adpted t simulate -D rugh surfaces []..1. Gemetric Mdeling f -D Gaussian Rugh Surfaces Fr a rugh surface f the size f L x L y, the prfile is frmed by means f the peridic extensin in the directins f x and y, respectively. Let fx, y be the height at any lcatin x, y, then use -D Furier extensin t act n it, i.e., fx, y = 1 iπm a mn exp x + iπn y 1 L x L y L m= n= x L y where a mn is cmplex amplitude f texture wave and Gaussian randm variable [3]. Let mx =πm/l x, ny =πn/l y. Due t Furier sum f Gaussian variable, fx, y als bviusly submits t Gaussian distributin. T btain surface crrelatin functin, we cnstruct the functins as fllws: fx 1,y 1 fx,y = 1 am1 n 1 a m n L x L y m 1 = n 1 = m = n = exp [ iπ L x m 1 x 1 m x + iπ L y n 1 y 1 n y ]

3 Prgress In Electrmagnetics Research B, Vl. 63, T satisfy the requirement f Gaussian pwer spectrum, fx 1,y 1 fx,y = σ Cx 1 x,y 1 y = x= y= d x d y exp i x x 1 x +i y y 1 y W G x, y 3 where σ is the RMS height, Cx 1 x,y 1 y the crrelatin functin which describes the cherence between different pints n the surface separated by the distance between the pint x 1,y 1 andx,y, and W G x, y the crrespnding pwer spectral density functin PDF f surface. Cmparing Eq. with Eq. 3, we find that the quadruple summatin series f Eq. can be assciated with Krnecer delta functin, then is simplified as duble summatin. Crrespndingly, spatial wave numbers in integral variables are discretized. The abve prcesses can be expressed as fllws [4, 5]: am1 n 1 a m n = δm1 m,n 1 n A m1 n 1 4 d x =Δ x = π, d y =Δ y = π L x L y 5 x = mδ x, y = nδ y 6 In this case, 1 A mn exp i x x 1 x +i y y 1 y L x L y m= n= = π L W G x, y expi x x 1 x +i y y 1 y x L y m= n= A mn = a mn a mn = a mn =π L x L y W G x, y 7 which gives the cnditin which mdule a mn satisfies. Hwever, fr cmplex cefficient a mn,next step will discuss the satisfied relatin between the real and imaginary parts f a mn. T generate a real sequence, the requirement fr F K x,k y isthatfk x,k y =F K x, K y andfk x, K y = F K x,k y. At the same time, a mn is inverse Furier transfrm f fx, y. S it hlds that a mn = a m n 8 Further, due t the effect f Krnecer delta functin in Eq. 4, let m = m 1 = m, n = n 1 = n, i.e., a mn a m n = a mn a mn =. 9 which describes the relatin cmplex cefficient a mn as fllws: [Reamn ] = [Ima mn ] 1 [Rea mn ]Ima mn ] = Namely, Rea mn andima mn are independent Gaussian randm variables, and their variances are bth a half f a mn. Additinally, Eq. 8 is cmbined with the peridic characteristics f a mn,and we btain the relatin a Nl Nl = a Nl Nl = a Nl Nl a Nl Nl R. 11 At this pint we btain the real and imaginary parts f a mn thrugh the Gaussian distributin functin which frms cmplex cefficient Gm, n. Eventually, cmplex amplitude a mn t satisfying the abve relatins can be expressed by: a mn mx, ny =Gm, nπ L x L y W G mx, ny +G m, nπ L x L y W G mx, ny 1 Thrugh Eq. 1, the height value f fx, y f rugh surface can be generated by the FFT methd.

4 9 Sun, Xie, and Zhang This paper adpts Gaussian crrelatin functin and Gaussian PDF t generate -D Gaussian rugh surfaces, i.e., Cτ x,τ y =σ exp τ x lx τ y ly 13 W G x, y = l xl y σ 4π exp xlx 4 y l y 4 14 where τ x and τ y describe the separatin between any tw pints alng the x and y directins. The crrelatin length f the surface prfiles is given by l x and l y. The surface is istrpic if l x = l y and anistrpic if l x l y. The pwer spectral density functin f the surface W G x, y is related t the crrelatin functin via a -D Furier transfrm. a b c Figure 1. The simulated -D Gaussian rugh surfaces with texture characteristics: a l x =1m, l y =m,σ =.6m. b l x =1m,l y =1m,σ =.6m. c l x =5m,l y =1m,σ =.6m. Texture angle is 3.

5 Prgress In Electrmagnetics Research B, Vl. 63, Realizatins f -D Gaussian Rugh Surfaces with Texture As this paper is aimed at the statistically anistrpic rugh surfaces, we empirically limited the rati f the l x /l y t be larger than 4, which guarantees that the surface can present a mre anistrpic characteristic. Texture angle φ is defined as the angle between x-axis psitive directin and texture directin. W G x, y can be btained thrugh angle rtatin actin f the rtatin matrix n Eq. 14, i.e., [ ] [ x cs φ sin φ = y sin φ cs φ ] [ ] x y Under this circumstance, the -D Gaussian rugh surface by the FFT methd als develps crrespnding angle rtatin. The size f the simulated rugh surface is 4 4 m, and there are 14 sampling pints in x and y directins. The simulated results are shwn in Figure 1. Amng them, a shws the simulatin result, in which the RMS height σ is.6 m, crrelatin length l x 1 m, l y m and texture angle φ 3.Based n the abve parameters, b changes l y int 1 m, and c changes l x int 5 m and l y int 1 m. Frm Figure 1, we clearly find that the textures frm b becme narrwer than that frm c, and the height variatins frm c becme much mre bvius than that frm b. 3. SCATTERING THEORY FROM ROUGH SURFACE BY SSA In SSA, the gemetrical cnfiguratin adpted t reslve the wave-scattering prblem frm the -D randmly rugh surface is illustrated in Figure, where we cnsider a rugh interface z = h r, with r =x, y, between tw hmgenus half-spaces with permittivity ε 1 upper half-space, z> and ε lwer half-space, z< [6, 7]. The time dependence is assumed t be exp iωt. θ i and θ s are, respectively, incident and scattering elevatin angles, and φ i and φ s are the incident and scattering azimuth angles, respectively. The incident wave vectr can be expressed as K i = q ẑ,where and q are hrizntal and vertical prjectins f the incident wave vectr, respectively. The scattered wave vectr is K s = + qẑ, where and q are apprpriate cmpnents f the scattered wave vectr, respectively. q and q canbeexpressedasq = ω /c, q = ω /c,imq, q [8]. The unit vectr in the directin f incidence is: ˆK i =sinθ i cs φ iˆx +sinθ i sin φ i ŷ cs θ i ẑ Figure. Gemetry cnfiguratin fr the wave scattering frm -D surface.

6 94 Sun, Xie, and Zhang and the incident wave vectr K i = K i ˆKi =, q. The incident field can be expressed as: [ ψ inc R=exp i K i R =exp i r ] iq z =exp[ ik i z cs θ i x sin θ i cs φ i y sin θ i sin φ i ] 17 where R = r,q = x, y, z and ψ inc is electric field E r magnetic field H depending n the plarizatin. The unit vectr in the directin f scattering is: ˆK s =sinθ s cs φ sˆx +sinθ s sin φ s ŷ +csθ s ẑ 18 and the scattered wave vectr is K s = K s ˆKs =,q. The scattered field can be expressed as: ψ sc x, y, z = exp i r, + iqz S d 19 Cnsidering the case f far-field apprximatin, the scattering amplitude matrix crrespnding t the SSA methd can be mdified as, qq 1/ d r [ S = Pinc q + q π exp i r ] + iq + q h r, B i, M, ĥ exp i r d 4 where B, is respnsible fr the first rder cntributin, and the fllwing integratin fr the secnd-rder cntributin. Mrever, P inc is the incident wave pwer received by the rugh surface and can be expressed as: P inc = ψ inc x, y, dxdy 1 and where S =, M, ĥ, = B, = [ ] S11 S 1, B = S 1 S h rexp i r [ ] B11 B 1, M = B 1 B, + B, +, +q + q B d r 3 [ ] M11 M 1, B M 1 M = [ B 11 B 1 B 1 B ] which describe mutual transfrmatins f the EM waves f different plarizatins. They are discussed in detail in [9]. Mrever, subscripts 1 and dente vertical and hrizntal plarizatins, respectively. The left-hand number represents the plarizatin mde f the receiving antenna, and the right-hand number represents that f the transmitting antenna. Superscript dentes secnd-rder Bragg s ernel. Fr cnvenience, this paper nly discusses c-plarizatins HH and VV. It can be cnfirmed that, in a general case, M, ; =. In view f this, the term related t the functin M in Eq. is prprtinal t the slpes f rughness rather than t the heights. This term prvides a crrectin t SSA-I. In this paper, this term is realized thrugh inverse Furier transfrm t reduce the cmputatinal time. Let us set b 11, ; ε 1,ε b, ; ε 1,ε =ε ε 1 =ε ε 1 ε 1 q + ε q 1 1 ε 1 q + ε q q + q 1 1 q + q 1 1 5

7 Prgress In Electrmagnetics Research B, Vl. 63, where Then B 11, B,, B 11 ; ; ε 1,ε, B ; ; ε 1,ε q 1 ω = ε 1 c, q q 1 = ε 1 ω c, = q = ω ε c ω ε c, = b 11 ; ε 1,ε ε 1 q q ε 6 7, ω = b ; ε 1,ε c 8, [ ε ε 1 = b 11 ; ε 1,ε ε 1 q + ε q 1 ε 1 q q + ε q 1 + q +ε 1 ε ε 1 q + ε q 1 q + q ε 1 Ks q + Ks q +q q q 1 q ] 9, [ ω ε ε 1 = b ; ε 1,ε c ε 1 q + ε q 1 ] + q + q + q 1 q Here, we chse medium 1 as air and medium as general dielectric rugh surface which means that the cmplex relative permittivity f the air is ε 1 =1,, and that f medium is calculated by using Debye frmulas [3]. In terms f rugh surface scattering amplitudes calculated by the SSA-II methd, the NRCS can be btained by, σpq, =4πqq ΔS pq ΔS pq 31 3 where ΔS pq,,, = S pq S pq 3 Expressin 31 represents the scattered field crrespnding t single rugh surface, and subscript pq dentes plarizatin state. Due t the randm characteristics f the rugh surface, the final bistatic NRCS and bacscattering cefficient are calculated as an average, i.e., σpq = σpq NUMERICAL RESULTS AND ANALYSIS In the fllwing, the radar frequency is 1.3 GHz. The size f the rugh surface is L x = L y =4m, sampled with 14 pints in each directin, i.e., the sample interval is λ/6. The relative permittivity f rugh surface is ε =4.1.98i. The statistical parameters f the simulated rugh surface are chsen as l x =1m,l y =1m,σ =.5 m. Each average NRCS is btained ver 5 realizatins f rugh surfaces. Fr simplicity, the fllwing scattering results, bth bacscattering and bistatic scattering, in this paper are all implemented in the incident plane, i.e., φ i =, φ s = fr bacscattering scattering, while φ i =, φ s = 18 fr bistatic scattering.

8 96 Sun, Xie, and Zhang 4.1. Cmparisn f NRCS between SSA-I and SSA-II The single bistatic and bacscattering NRCS versus angles frm the rugh surface fr the texture angle f is shwn in Figure 3. Amng them, a and b represent the bistatic scattering fr the incident angle f 45. c and d represent that fr the incident angle f. e and f represent the bacscattering cefficient versus incident angles. Frm Figure 3a Figure 3d, it is seen that fr bistatic case, the NRCS frm the SSA-II methd is slightly larger than the SSA-I methd fr scattering angles departing frm specular directins. Fr incident angle θ i =45, the distinctin mainly appears at negative scattering angles, whereas fr θ i =, difference exists fr bth bacward and frward directins. Hwever, near the specular directin, such differences fr bth methds are very minr. Frm Figure 3e and Figure 3f, it is seen that fr the bacscattering case, in the quasi-specular regin, the bacscattering cefficients fr bth methds are almst the same, but as the incident angle increases, the cefficient fr the SSA-II methd is larger than that fr the SSA-I methd. It is wrthy t pint ut that the SSA-II methd spends mre cmputatinal time than the SSA-I methd, but calculatin accuracy f the SSA-II methd is bviusly higher than that f the SSA-I methd. S this paper adpts the SSA-II methd t study the scattering characteristics f the rugh surface with different textures. Bistatic NRCS db HH plrizatin SSA-I SSA-II Scattering angle a Bistatic NRCS db VV plrizatin SSA-I SSA-II Scattering angle b Bistatic NRCS db HH plrizatin SSA-I SSA-II Scattering angle c Bistatic NRCS db VV plrizatin SSA-I SSA-II Scattering angle d

9 Prgress In Electrmagnetics Research B, Vl. 63, Bacscattering cefficient db HH plarizatin SSA-I SSA-II Bacscattering cefficient db VV plarizatin SSA-I SSA-II Incident angle e Incident angle f Figure 3. C-plarizatin scattering frm a single rugh surface: a d shw the variatin f bistatic cefficients with scattering angles. e and f give the dependence f bacscattering cefficients n incident angles. And blac lines represent SSA-I and green lines represent SSA-II. 4.. Cmparisn f Average NRCS fr Rugh Surface with Different Textures by SSA-II The average bistatic and bacscattering NRCS versus angles frm the rugh surface with different texture angles in SSA-II is shwn in Figure 4. Amng them, a and b represent the bistatic NRCS fr the incident angle f 45, and c and d represent the bacscattering cefficients. Frm Figure 4a and Figure 4b, it is seen that fr bistatic case, as the texture angles increases frm t 9, the NRCS frm the large texture angle is significantly larger than that frm the small ne fr scattering angles departing frm specular directins. Hwever, near the specular directin, such differences fr different texture angles are very minr. In additin, it is wrthy t pint ut that fr the texture angle f 9, the NRCS near the bacward regin significantly increases, especially in VV plarizatin. This is because the incident plane is perpendicular t the texture directin, and mre scattering facets cntribute t the bacward regin. Frm Figure 4c and Figure 4d, it is seen that, fr the bacscattering case, in the quasi-specular regin, the bacscattering cefficients fr different texture angles are almst the same, but as the incident angle increases, the cefficient fr the large texture angle is larger than that fr the small ne. Average bistatic NRCS db 1 HH plarizatin φ = φ = 45-3 φ = Scattering angle a Average bistatic NRCS db 1 VV plarizatin φ = φ = 45-3 φ = Scattering angle b

10 98 Sun, Xie, and Zhang Average acscattering cefficient db 1 5 HH plarizatin φ = φ = 45 5 φ = Incident angle c Average acscattering cefficient db d VV plarizatin φ = φ = 45 φ = Incident angle Figure 4. Average c-plarizatin scattering with different texture angles, i.e., φ =, 45, 9 : a and b shw the variatin f bistatic cefficients with scattering angles. c and d give the dependence f bacscattering cefficients n incident angles Effects f Statistical Parameters n Average Bacscattering Cefficients In the fllwing, we merely chse the incident angle f 3. Accrding t different crrelatin lengths and RMS heights, the calculatins are divided int fur grups which are shwn in Table 1. Table 1. Statistical parameters f different textures. Case Crrelatin length Crrelatin length in x directin in y directin RMS height 1 1 m m 1m 1 m 1m 1m 3 5m 1m 1m 4 1 m 1m.5 m Average bacscattering cefficient db HH plarizatin case1 case case3 case4 Average bacscattering cefficient db 5 VV plarizatin case1 case case3 case Texture angle a Texture angle Figure 5. Average bacscattering cefficients versus texture angles. a dentes HH plarizatin, and b dentes VV plarizatin. b

11 Prgress In Electrmagnetics Research B, Vl. 63, The calculated results are shwn in Figure 5. It is seen that fr the texture angle f 9,the bacscattering cefficients in any case are basically maximum and begin t reduce gradually fr the texture angles departing frm 9. Cmparing case with case 4, we find that the bacscattering cefficients frm case are significantly greater than that frm case 4 except specular part. This is mainly because the RMS height f case is tw times greater than that f case 4. Namely, the texture surface crrespnding t case is much rugher than that f case 4. Mrever, cmparing case 1 with case 3, under the cnditin that the rati f crrelatin length l x t l y is 5 t 1 fr bth cases, the bacscattering cefficients frm case 3 is far larger than that frm case 1 except specular part, which fully illustrates that small crrelatin length has a significant impact n the bacscattering. It is wrthy t pint ut that because the SSA-II methd is related t the slpe f surface height, the bacscattering cefficients are nt strictly symmetric abut the texture angle f 9, which is different frm the results f the SSA-I methd Cmparisns f Experimental Data with Results Calculated by SSA-II The average bacscattering cefficients calculated by the SSA-II methd are cmpared with the experimental data frm asphalt surface in the literature [31], Ulaby et al., 1986, Chapter 1, Figure 1.8. Amng them, the size f the expnent spectrum rugh surface is L x = L y =15λ inc, where λ inc is the EM wavelength. The statistical parameters are l x = l y =3.574 mm, σ =1.44 mm. The calculated sample number is 1. The calculated results are shwn in Figure 6. Fr Figure 6a, the incident frequency is f =8.6GHz, and the relative permittivity f medium is ε r =5,, while fr Figure 6b, f =17GHzandε r =9,. Average bacscattering cefficient db Frequency = 8.6 GHz ε r = 5, SSA-II_HH SSA-II_VV Exp.HH Exp.VV Incident angle a Average bacscattering cefficient db Frequency = 17 GHz ε = 9, r SSA-II_HH SSA-II_VV Exp.HH Exp.VV Incident angle b Figure 6. Cmparisn f results by the SSA-II methd and measured data f an asphalt surface at a 8.6 GHz and b 17 GHz, respectively. Frm Figure 6, it is seen that the results calculated by the SSA-II methd are in gd agreement with the experimental data at whether high frequencies r lw frequencies. Mrever, it is pinted ut that the theretical values are slightly larger than the measured data. This difference may be caused by the discrepancy f the actual rad surface and generated surface, which is revealed by the increasing gap with increasing incident frequency. Hwever, within the range f allwable errr, the SSA-II methd is very effective in the EM calculatin f rugh surfaces.

12 3 Sun, Xie, and Zhang 5. CONCLUSION In this paper, the SSA-II methd is applied t calculate the scattering frm -D randmly Gaussian rugh dielectric surfaces fr different texture mdels. A cmparative study has been dne n the distinct prperties f bth NRCS due t texture effects amng waves. Frm the numerical results f bistatic and bacscattering NRCS, it is seen that the differences between the SSA-I and SSA-II methds are revealed, and the effects f statistical parameters and texture angle n the scattering are further analyzed. In summary, the analysis presented in this paper helps t establish better understanding f the scattering features f the -D texture rugh surface. Meanwhile, this paper prvides imprtant reference value fr the texture infrmatin f -D rugh surface. It is wrthy t pint ut that the numerical scattering results f texture rugh surfaces remain t be further verified thrugh measurement data. ACKNOWLEDGMENT The authrs wuld lie t than the Fundamental Research Funds fr the Central Universities, the Natinal Natural Science Fundatin f China, and the Aerspace Research Institute f Materials and Prcessing Technlgy fr the supprt t this research. REFERENCES 1. Tsang, L. and J. A. Kng, Scattering f Electrmagnetic Waves, Advanced Tpics, Wiley Series in Remte Sensing, Wiley Interscience, New Yr, 1.. Zha, Y. W., M. Zhang, X. Geng, and P. Zhu, A cmprehensive facet mdel fr bistatic SAR imagery f dynamic cean scene, Prgress In Electrmagnetics Research, Vl. 13, , Mcdaniel, S. T., An extensin f the small-slpe apprximatin fr rugh surface scattering, Waves in Randm Media, Vl. 5, N., 1 14, Sun, R. Q., G. Lu, M. Zhang, and C. Wang, Electrmagnetic scattering mdel f the Kelvin wae and turbulent wae by a mving ship, Waves in Randm Media, Vl. 1, N. 3, 51 54, Chen, H., M. Zhang, and H.-C. Yin, Facet-based treatment n micrwave bistatic scattering f three-dimensinal sea surface with electrically large ship, Prgress In Electrmagnetics Research, Vl. 13, , Warnic, K. F. and W. C. Chew, Numerical simulatin methds fr rugh surface scattering, Waves in Randm Media, Vl. 11, N. 1, R1 R3, Thrss, E. I., The validity f the Kirchhff apprximatin fr rugh surface scattering using a Gaussian rughness spectrum, J.Acust.Sc.Am., Vl. 83, N. 1, 78 9, Gu, L.-X., Y. Liang, J. Li, and Z.-S. Wu, A high rder integral SPM fr the cnducting rugh surface scattering with the tapered wave incidence TE case, Prgress In Electrmagnetics Research, Vl. 114, , Lee, P. H. Y., et al., Wind-speed dependence f small-grazing-angle micrwave bacscatter frm sea surfaces, IEEE Trans. Antennas Prpagat., Vl. 44, N. 3, , Sajjad, N., A. Khenchaf, et al., An imprved tw-scale mdel fr the cean surface bistatic scattering, Prc. Of Internatinal Gescience and Remte Sensing Sympsium, Vl. 1, , Vaitilingm, L. and A. Khenchaf, Radar crss sectins f sea and grund clutter estimated by tw scale mdel and small slpe apprximatin in Hf-VHF bands, Prgress In Electrmagnetics Research B, Vl. 9, , Vrnvich, A. G., Small-slpe apprximatin in wave scattering by rugh surfaces, Sv. Phys.- JETP, Vl. 6, 65 7, Berginc, G. and C. Burrely, The small-slpe apprximatin methd applied t a threedimensinal slab with rugh bundaries, Prgress In Electrmagnetics Research, Vl. 73, , 7.

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