Mahamadou Tembely, Matthew N. O. Sadiku, Sarhan M. Musa, John O. Attia, and Pamela Obiomon

Transcription

1 Journal of Multidisciplinar Engineering Science and Technolog (JMEST) Electromagnetic Scattering B Random Two- Dimensional Rough Surface Using The Joint Probabilit Distribution Function And Monte Carlo Integration Transformation Mahamadou Tembel, Matthew N. O. Sadiku, Sarhan M. Musa, John O. Attia, and Pamela Obiomon Ro G. Perr College of Engineering Prairie View A&M Universit Prairie View, TX mtembel@student.pvmau.edu; sadiku@ieee.org; smmusa@pvamu.edu; joattia@pvamu.edu; phobiomon@pvamu.edu Abstract Electromagnetic scattering b two dimensional random rough surface is studied. After generating a two-dimensional rough surface, the joint probabilit densit function of the surface height and slope is computed and used to determine the electromagnetic scattering field. The ra-tracing base model is utilized. Also Monte-Carlo method is emploed to transform an infinite to a finite integration. In this paper two case studies have been performed and compared with each other. For the first case, we worked on the suggested method on [1] to find the D electromagnetic scattering field. In this, the authors for their future work have proposed a wa to solve the D scattering problem. Their method consisted of computing the PDF of the scattering field in and direction and use them to determine the scattered field. For second case we discuss our proposed method which is to use the joint PDF of surface height and slope instead of the one proposed in [1] to determine the D scattering field. The reason for using the joint PDF of the surface height and slope is that an point on a surface can be onl well represented b its height and slope. Both methods are compared each another. And the two methods are then compared to others works. From these comparisons we have shown that our method gives better result. Kewords Electromagnetic scattering, Rough surface, Probabilit densit function, Monte Carlo, Scattering coefficient. I. INTRODUCTION The scattering of electromagnetic b twodimensional random rough surfaces have been developed in the last centur. Nowadas researchers interested in solving the problem have increased because of the impact of this phenomenon in diverse disciplines of sciences. These include communication, geophsics and remote sensing of the earth. The classic methods applied to two-dimensional random rough surface scattering are the small perturbation and Kirchhoff approimation [], [3]. These two method are limited in their domain of validit. Monte Carlo simulations of direct solutions of electromagnetic scattering problem is popular due to advent of modern computers and the advancement of fast numerical methods. Monte Carlo is also restricted because it requires more powerful numerical approach and the use of long surface. In this paper, the electromagnetic scattering field is studied using the joint PDF of height and slope. We assume that the surface height and slope are uncorrelated. First, we have generated a D random rough surface using MATLAB. Ra tracing technique is emploed to investigate the electromagnetic field [4], [5]. Second, referring to the two-dimensional configuration in polar coordinates (see figure 4) the D surface slope is calculated. After getting the surface slope we determine the joint PDF of the slope and the height where the PDF of the height is calculated too. Third, this Joint PDF has been utilized to compute the scattering filed. The scattering coefficient is plotted against the scattering angle in MATLAB. And our result is compared with others works (see figure 6). These include Kirchhoff approimation and Monte Carlo method [1], [16]. II. D ROUGH SURFACE GENERATION In this section we generated a two-dimensional random rough surface. Several methods have been developed to generate a random rough surface. These include convolution method, lines-oriented method for generating inhomogeneous random rough surface and directed Fourier transform [6], [7], [8]. In this paper, the rough surface parameters are derived analticall b using convolution method for its generation. A. D Rough Surface Characteristics JMESTN

2 In general a surface is characterized b its spectrum densit, correlation length autocorrelation function and its autocovariance. The Gaussian height probabilit densit function is given b [9], [10]. P(Z = f(, )) = 1 (Z=f(,)) e ρ (1) ρ π Journal of Multidisciplinar Engineering Science and Technolog (JMEST) F γ γ = N 1 N 1 n =0 n =0 f n n e jπ(n γ N + n γ N ) with γ = 0, 1,.. N 1 γ = 0, 1,.. N 1 (7) where f(, ) represent the height function of the D random rough surface and ρ its mean square. For D surface the spectrum densit function W(K) is defined as follows: W(K) = ( cl ρ 4π ) e Kcl 4 () where cl is the correlation length. K the spatial angular frequenc vector, is defined b: K = (K, K ), K = K + K (3) N and N represent the numbers of arras in and direction respectivel for D. Therefore we have the surface height f n n = F γ γ e jπ (n γ N 1 N 1 N γ =0 γ =0 +n γ N ) with n = 0, 1,.. N 1 n = 0, 1,.. N 1 (8) From the spectrum densit function we determine auto-correlation function (ACF) + ACF = W(K)e ik(+) dk = h cl e cl where h is the rms height and cl and cl are the correlation lengths in and respectivel. As the ACF and the height are both Gaussian distributions then the slope will be also Gaussian. The slope in and give S = f and S = f (4) (5) We use MATLAB to perform the generation of the D surface (see figure 1). Parameters Table 1. D rough surface parameters values Number of surface points rllength of surface h - rms height values 00 1λ 0.1 λ 0.08 λ Cl and cl - correlation length ( in and ) Then we get the power spectrum densit function of the surface. P S (S, S ) = where w = ρ cl Gaussian rough surface. S 1 w e S w πw w represents the rms slope for (6) B. Algorithm and Plot for D Rough Surface Generated In convolution method the D discrete Fourier transform (DFT) is given b F = DFT(f) then the inverse DFT is f = DFT 1 (f) Figure 1. Two-dimensional rough surface generated. JMESTN

3 Journal of Multidisciplinar Engineering Science and Technolog (JMEST) P D (θ s ) = P h (θ s ) P h (θ s ) (10) We need to mention that in [1] the D rough surface case has not been studied but onl 1D rough surface has been done. The authors have just proposed a method to solve the D case. In this paper we have studied their proposed method and computed the scattering field [1] E s (θ s ) = E 0 F s (θ i, θ s, n) P D (θ s ) e p( jφ(z, θ i )) dz (11) Figure. Probabilit distribution function (PDF) of D rough surface height. III. THE SCATTERING FIELDS COMPUTATION AND PLOTS In this part of this paper we report our work on two proposed methods. We stud first the proposed method in [1]. Second our proposed method is studied and these two methods are compared with each other. Last, both methods are compared with others works eisting the literature. A. Case Stud 1 Using P D (θ s ) for Scattering Field Computation Here F s (θ i, θ s, n) is Fresnel coefficient in function of incident angle θ i, scattering angle θ s and inde of refraction n Now let us appl Monte Carlo method to transform infinite integration to finite. Then the integral becomes e p( jφ(z, θ i )) P(Z) dz = e p( jφ(z, θ i)) P(Z) dz P(Z) Therefore = 1 k e jφ k E s (θ s ) = E 0 F s (θ i, θ s, n) P D (θ s ) 1 k k e jφ k 1 (1) Thus the electromagnetic scattering field for D is E s (θ s ) = E 0 F s (θ i, n) P h (θ s ) P h (θ s ) 1 k e jφ k k 1 k 1 Here we worked on the proposed method described in [1] to calculate the electromagnetic scattering field. In their paper the authors have worked on the 1D surface problem where the probabilit densit function of the scattering field in θ s direction has been used to calculate the electromagnetic scattering. The PDF is given b P 1D (θ s ) = cl σ h cos (α) π ep ( (tan(α) cl σ h )) (9) where tan(α) represents the surface inclination. For the stud of D rough surfaces, the mentioned that there is no a simple relationship that links the local normal orientation and the scattering direction. Therefore for the future work on D case, the proposed to use the PDF of the scattering field in and direction where the orientations of all the microfacets of rough surface are computed. As the two PDFS are independent then the joint probabilit is given b Figure 3. Scattering coefficients for case stud 1 using P D (θ s ). B. Case Stud Using P sz (S, Z) for Scattering Field Computation For our stud of electromagnetic scattering b two-dimensional rough surface we propose to use the PDF of the D surface height and slope. This is used because an point of the surface is described b its JMESTN

4 Journal of Multidisciplinar Engineering Science and Technolog (JMEST) height Z and slope S. That shows the surface is well represented b its slope and height. Therefore we believe here that the use of P sz (S, Z) to calculate the scattering field would be more suitable and appropriate compared to the one proposed in [1] ( ie P D (θ s )). E s (θ s ) = E 0 F s (θ i, θ s, n) P sz (S, Z) 1 k e jφ k k 1 Figure 4. D surface configuration in polar coordinates. Figure 5. Comparison scattering coefficients for Case stud 1, using P sz (S, Z) and P D (θ s ). It assumes that the surface height and slope are uncorrelated. There are two independent Gaussian variables and the probabilit densit function PDF [11] P sz (S, Z) = P s (S) P z (Z) = 1 From Figure 4, we have S e σ Z s σ z πσ s σ z (13) S = S cos + S sin with S and S represent the surface slopes in the (O), (O) direction (see equation 5). According to (O), is the azimuthal direction. The covariance matri: Cov = [ E(Z ) E(ZS) E(SZ) E(S ) ], where for eample E(Z ) represent the epected value of the surface height. Assume that the surface height and slope are uncorrelated then E(ZS) = E(SZ) = 0 where E(Z ) = σ z and E(S ) = E((S cos + S sin ) ) = (ω cos ) + (ω sin ) + E(S S sin cos ) As S and S are uncorrelated therefore E(S S ) = 0 Thus σ s = (ω cos ) + (ω sin ) with ω and ω are the variance of the slope in the O and O. Therefore the matri becomes: Cov = [ σ z 0 0 σ s ] Figure 6. NRCS of a Gaussian rough surface and area of 66λ 66λ A =λ/0. ( h rms, l c ) = (0. 16λ, 0. 95λ), θ i = 30, ε r = 4 j, fc = 0.3 GHz and r a = 7.69λ, averaged over two realizations using DD-FDTD with 5 X 5 subdomain, Le = 3λ [15]. JMESTN

5 C. Results Comparison Figure 5 shows the comparison of scattering coefficient for case stud using P sz (S, Z) and that of P D (θ s ). Both cases studied have been realized on D rough surface with the same surface characteristics and parameters defined on Table 1. For our case stud of P sz (S, Z), we can notice from the plot that the scattering field increase rapidl with scattering angle and at around 80 degree it got an inflection point. From that point it starts increasing b 0dB until it reaches a critical point 100 degree scattering angle where it starts decreasing. In contrast to our method, the proposed method for case stud using P D (θ s ) the scattering field is almost constant( 0dB) and at 80 degree of scattering angle it gains around 10dB before it decrease b around 5dB and from that is almost constant (5dB) again. There is an agreement between our method and those presented in the literature. These include Monte Carlo Simulation [13], Kirchhoff Approimation [14] and the method use in [15] (see figure 6). This has demonstrated the accurac and effectiveness of our method. IV. CONCLUSION In this paper we have proposed a new method to compute the electromagnetic scattering field using the joint probabilit distribution function of the height and slope of D rough surface. This method has been compared with the proposed method in [1] for the future D stud b plotting their scattering coefficient. And the contrast between the two methods can be clearl notified. But our method is more accurate because it has shown its agreement with others works in the literature. electromagnetic wave. REFERENCES [1] Y. Cocheril and R. Vauzelle A new Ra- Tracing based wave propagation model including rough surfaces scattering Progress In Electromagnetics Research, PIER 75, pp , 007 [] L. Tsang, C.H. Chan and K. Pak Backscattering enhancement of a twodimensional random rough surface( threedimensional scattering) based on Monte Carlo simulations, Journal of the Optical Societ of America A Vol.11, Issue, pp (1994) [3] C. H. Chan, L. Tsang and Q. Li Monte Carlo Simulations of Large-Scale one Dimensional Random- Surface Scattering at Near-Grazing Incident: Penetrable Case IEEE Transactions on Antennas and Propagation vol.46. No.1, Januar 1998, pp Journal of Multidisciplinar Engineering Science and Technolog (JMEST) [4]] Louza, S., Electromagnetic Fields Scattered b Rough Surfaces. Ph.D. Dissertation, Universit of Alabama in Huntsville, Alabama [5] Bergstrom, J. Powell, A. F. Kaplan A ratracing analsis of the absorption of light b smooth and rough metal surface Journal of Applied Phsics vol.101, Issue 11, pp (007) [6] M. Tembel, M.N.O. Sadiku and S. M. Musa Markov Chain Analsis Of Electromagnetic Scattering B A Random One-Dimensional Rough Surface Journal of Multidisciplinar Engineering Science and Technolog (JMEST), Vol. 3 Issue 7, Jul 016, pp [7] K. Uchida, J. Honda and K.Y. Yoon An algorithm for rough surface generation with inhomogeneous parameters International Conference Processing Workshops 009 [8] L. Tsang, J. A. Kong, K.H. Ding, C.O. Ao Scattering of electromagnetic waves; Numerical Simulation. John Wile & Sons, 001, pp [9] F. Gebali, Analsis of computer and communication network. New York: Springer 008, pp [10] G. J. Anders, Probabilit Concepts in Electric Power Sstems. New York: John Wile & Sons, 1990, pp [11] C. Bourlier and G. Berginc Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian and sea correlations Institute of phsics publishing Waves Random Media 13 (003), pp [1] M. N. O. Sadiku, Monte Carlo methods for electromagnetics. Boca Raton London New York: CRC Press Talor & Francis, April 9, 009. [13] J. T. Johnson, L. Tsang, T. Shin, K. Pak, C.H. Chan and Y. Kuga Backscattering Enhancement of waves from two-dimensional perfectl conducting random rough surfaces: A comparison of Monte Carlo Simulation with eperimental data IEEE Transactions on Antennas and Propagation, vol. 44, no. 5, Ma 1996, pp.748. [14] F. Shi, W. Choi, M.J.S Lowe, E. A. Skelton and R.V. Craster The validit of Kirchhoff theor for scattering of elastic waves from JMESTN

6 rough surfaces Published b Roal Societ, 6 Ma 015. Journal of Multidisciplinar Engineering Science and Technolog (JMEST) [15] Z.H. Lai, J. F.Kiang and R. Mittra A Domain Decomposition Finite Difference Time Domain (FDTD) Method for Scattering Problem from ver Large Rough Surfaces. IEEE Transactions on Antennas and Propagation, vol. 63, no. 10, October 015, pp [16] A. E. Shadare, M. N.O Sadiku, S. M Musa Markov Chain Monte Carlo Analsis of Microstrip Line International Journal of Enngineering Research and Advanced Technolog, vol. 0 Issue. 03, March-016 JMESTN