Method of moments applied to the analysis of rough surfaces modelled by fractals

European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2000 Barcelona, 11-14 September 2000  ECCOMAS 1 METHOD OF MOMENTS APPLIED TO THE ANALYSIS OF ROUGH SURFACES MODELLED BY FRACTALS Mercè Vall-llossera, Nuria Duffo, Adriano Camps, Ignasi Corbella, Javier Bará, Francisco Torres, Miquel Guillamont Department of Signal Theory and Communication Campus Nord UPC Edifici D-3 Jordi Girona, 1-3, 08034 Barcelona, Spain E-mail: m[email protected], tel: 34-934017261, Fax: 34-4017232 Key words: Method of Moments, Scattering, Rough surfaces, Fractals. Abstract. The Scattering and Emissivity of rough surfaces involve solutions to non-linear differential equations. Different approaches have been used in the literature to obtain approximate solutions under some hypothesis. For example Kirchhoff solution is used when the roughness is gentle on the scale of the wavelength. In this paper the Method of Moments is used to analyze the scattering of arbitrary surfaces. No approximation about the scale roughness is necessary. Both Gaussian and Fractal surfaces have been modeled and compared. The introduction of fractal geometry provides a new tool to describe natural rough surfaces. A first inside to the properties and parameters that describe fractal geometry has been done in order to characterize them statistically. It has been demonstrated that geometrical and scattering characteristics are controlled by Fractal descriptors, including fractal dimension1. As a first step, our simulations refer to a (topological) one-dimensional (1-D) profile embedded in a two-dimensional (2-D) space. Physically, this corresponds to assume that both the electromagnetic field and the surface height are constant along a fixed direction. Extension to the case of a 2-D surface embedded in a three-dimensional (3-D) space is not conceptually difficult, but any simulation run requires a much longer computational time. Furthermore, scattering results obtained for 1-D profiles give also a good indication of scattering dependence on 2-D surface parameters. M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 2 1 INTRODUCTION The problem of solving electromagnetic wave scattering and emission from randomly rough surfaces is of great interest in remote sensing and telecommunication applications. Both analytical and numerical methods have been developed for its evaluation. Among the various theories, the most well known ones are the Small Perturbation Method (SPM), Kirchhoff approximation (tangent plane approximation) and Geometric Optics (GO). The validity of these analytical methods depends on the approximation made pertaining to the physical conditions such as the surface roughness and frequency. For surfaces with roughness scale small compared with the wavelength, the method of Kirchhoff is used, while the traditional perturbation method applies when rms height and rms slope of the rough surface are small relative to the wavelength. In either case, additional assumptions must be done to obtain mathematically tractable solutions. Recently, new approaches to the scattering problem have emerged in order to extend the applicability of these models into the intermediate frequency range. The Integral Equation Method (IEM) seeks a correction term to the surface current in addition to the Kirchhoff approximation and the Phase Perturbation Method perturbs only the field amplitude. The verification of theories requires experimental measurements of surface scattering coefficients under controlled laboratory. Nowadays, computer simulation offers an efficient alternative. Axline and Fung2 simulated the wave scattering from a perfectly conducting random surface by calculating the surface current density induced by a impinging plane wave by the Method of Moments (MoM). Chen and Bay3 extended this simulation technique to include backscattering from dielectric surfaces. On the other hand, the characterization of wave interaction with rough surfaces, such as sea surfaces, ocean bottoms and rough terrain needs of mathematical models of such rough surfaces. The introduction of fractal geometry provides a new tool to describe naturally occurring rough structures, since fractals hold in balance long-range order and short-range disorder and can be used to describe both deterministic and random structures or an appropriate blend. In this paper we use fractal function to model rough surfaces. First of all, we present a suitable fractal model proposed by Jaggard and Sun1. Relations between its defining parameters or fractal descriptors and those of traditional random model are enclosed. Section 3 is devoted to the analysis of random surfaces using MoM. Section 4 demonstrates that HH and VV scattering coefficients obtained from fractal surfaces are very similar to those from gaussian surfaces with the same statistics. 2 FRACTAL MODEL In this first study the fractal model proposed by Jaggard and Sun1 is used. They proposed a zero-mean, band-limited fractal function, expressed as a weighted sum of periodic functions: () ∑ − = +−= 1 0 0)(1)( N n n n nxbKsinDCxfr φσ (1) M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 3 Where, D is the fractal dimension in the function and gives a measure of the surface roughness, ranging from D=1 (smooth periodic curve) to D=2 (rough, area filling curve). K0 is the fundamental spatial wave number ( e K λπ /·2 0=, in which λe is defined as the fundamental spatial wavelength), b (>1) is the spatial frequency scaling parameter, φ n are arbitrary phases and N is the number of tones. The amplitude control factor () [] () [] [] 2/1 2 2/1 2 2 )1(1 )2(2 11 112           −− − =           −− −− =NN D DD D D C (2) is chosen so that the function has a standard deviation (rms height) σ, while the value of b can be chosen such that the fractal function is almost periodic. Obviously, the periodic functions of increasing frequency in the summation of equation (1) produce the fine structures. Clearly, other periodic functions could be used in (1) to replace the sine function if it was desired. This function has a finite band of spatial frequency and exhibits self-similarity over the corresponding finite range of resolution. 2.1 Relations between fractal and traditional parameters As we can see from relation (1), the structural profile of the rough surface is determined by the parameters σ (rms height), D (fractal dimension), b (frequency scaling), K0 (fundamental wave number), and N (number of tones). The traditional parameters used in random surface modeling are σ (rms height), Γ (correlation length), and σs (rms slope). In order to compare gaussian surfaces and fractal ones with the same random parameters it is necessary to relate these two sets of parameters. The only common parameter is σ (rms height). The rms slope of this kind of fractal surfaces can be found by deriving the rms value of the first derivative of function (1), which results in: () [] () [] [] [] 2/1 22 22 2 2 0)1(1 )1(1 11 11           −− −− −− −− = Db Db D D K NN N s σσ (3) Note the special case σ s=K0 · σ when either D=1 or N=1. Figure 1 graphically shows the dependence of σ s with the fractal parameters (D, N, b and λ e). It is demonstrated that increasing D increases always the rms slope, and this increment becomes much faster for D greater than 1.4. On the other hand, plots in figure 1 make evident that higher values of b, σ , and N obtain higher values of rms slope; meanwhile contrary behavior is observed varying λ e (see figure 1c)). The correlation length of this model can be found with the aid of the autocorrelation coefficient ρ ( τ ) of the fractal function given by: M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 4 () [ ] () [] () ∑ − = − −− −− = + =1 0 0 2 2 2 )cos(1 11 11 )(),( )(),( )( N n n n N rr rr bKD D D xfxf xfxf τ τ τρ (4) The angle brackets denote the ensemble average. The correlation length Γ is defined as the first root of ρ ( τ )=1/e when τ increases from zero. Figure 2 shows the variation of the rms slope (fig. 2 a)) and the correlation length (fig. 2b)) respect to the fractal dimension. These plots have been obtained for f=5GHz, λ e =2· λ 0=12 cm, N=6 and σ =0.35. It is observed that the equivalent height deviation for the fractal surface ( σ =0.35) equals to the gaussian one for D=1. Furthermore, figure 2a) shows a maximum for D=1.4 (the same value where the behavior of the rms respect to D changes). Finally, figure 2b) demonstrates that the equivalent correlation length Γ decreases as D increases, which confirms that the fractal dimension is a measure of the roughness. But no clear dependence is observed with b. σ s = f(N, b, σ, λe) N = 6 σ = 0.15 cm λe = 3 λ0 λ0 = 3 cm σ s = f(N, b, σ, λe) N = 6 b= 2e/3 λe = 3 λ0 λ0 = 3 cm σ = 0.10 σ = 0.15 σ = 0.20 σ = 0.30 σ s = f(N, b, σ, λe) N = 6 b= 2e/3 σ = 0.20 λ0 = 3 cm λe = 8 λ0 λe = 5 λ0 λe = 3 λ0 λe = λ0 σ s = f(N, b, σ, λe) b= 2e/3 σ = 0.20 λe = 4 λ0 λ0 = 3 cm a) c) d) b) Figure 1: rms slope respect to fractal dimension varying other fractal parameters: a) Different values b, b) Different values of σ , c) Different values of λ e , and d) Different values of N . M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 5 Figure 2: a) σ respect to the fractal dimension (D), b) Γ respect to fractal dimension (D). 3 METHOD OF MOMENTS (MOM) Figure 3: The geometry of rough surface scattering in two-dimensional space. The problem of solving electromagnetic wave scattering from randomly rough surfaces presented in figure 3 has been solved by the Method of Moments. A version of the algorithm presented by Chen and Bay3 for computing the backscattering and bistatic scattering from rough dielectric surfaces has been implemented. As the algorithm formulation is detailed in the Chen and Bay3 paper we are not including it. It has already been mentioned in the introduction that this study refers to a (topologically) one-dimensional profile embedded in a two dimensional space. Physically, this corresponds to assume that both the electromagnetic field and the surface height are constant along a fixed direction. The reason of solving this problem instead of a 2-D surface embedded in a threedimensional space is that any simulation run requires a much longer computational time. To perform computer simulation a randomly rough curve with prescribed surface height density distribution and autocorrelation function has to be generated, as it is shown in figure 3. Gaussian curves have been generated using the method proposed by Fung and Chen4, a) b) x zEi Hi θ Ei Hi θ HH VV L L ∆x M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 6 meanwhile the fractal ones have been generated as it is described in this paper at section 2. In practice, a Gaussian taper function of the form exp(-g-2x2cos2 θ ) is applied to the incident field to improve the continuity at the edge of the illuminated area. Owing to finite computer storage and practical restrictions on the matrix size, the illuminated length (L in figure 3), must be finite. Repeated calculations of Ns segments (each of length L) must be performed to achieve meaningful estimates of the scattering coefficients. Axline and Fung2 make a discussion around the appropriate values for L, Ns and g parameters. 4 SIMULATIONS RESULTS The first step was to assess that MoM agrees with Kirchhoff solution under the hypothesis of validity of this last analytical method. It was created a gaussian correlated surface with roughness parameters K· σ =1.256, K·l= 8.38, σ s=0.21 and dielectric constant ε r=80. A segment of that surface is presented in figure 4a). Besides figure 4b) compares plots for the HH and VV backscattering coefficients obtained using MoM with results obtained using Kirchhoff approximation. In order to assure enough accuracy in using MoM 75 segments where created and analyzed. Everyone had a size twelve times the correlation length. After that, simulations comparing fractal and gaussian surfaces where carried out. Random fractal surfaces using the formulation presented in section 2 have been obtained and analyzed and the scattering coefficients have been compared with the ones obtained from gaussian surfaces with the same statistical parameters. These simulations demonstrated that traditional parameters for random surfaces could be obtained from fractal parameters in the way described in section 2. Figure 5a) plots two surface samples one gaussian (red dashed line) and another fractal (blue continuous line). Both with the same traditional parameters: a) b) Backscattering coefficient (dB) Figura 4: Figure 4: a) Gaussian profile with roughness parameters K· σ =1.256, K·l= 8.38, σ s=0.21 and dielectric constant ε r=80. b) Comparison of the backscattering coefficients between MoM simulation and Kirchhoff solution. M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 7 kσ=0.29, kl=3.64 and σs=0.11. The fractal profile has been created applying equation 1 with the following parameters: N=6, b=e/1.8, D=1.6, λe=4λ0. The roughness of those surfaces is very similar. In the simulation it has been used λ0=3cm and εr=16. HH and VV Backscattering coefficient for these two profiles are compared in figure 5. Good agreement is obtained. In order to achieve meaningful estimates of the scattering coefficients 85 segments have been analyzed using MoM in both kind of surfaces. Figura 5:a) Fractal profile (blue continuous line) with fractal parameters: N=6, b=e/1.8, D=1.6, λe=4λ0, σ=0.15cm, σs=0.12 compared with gaussian profile whose statistical parameters are kσ=0.29, kl=3.64 and σs=0.11. It has been used: λ0=3cm and εr=16. in both cases b) Comparison between HH and VV backscattering coefficients of a fractal surface and a gaussian one with the same traditional parameter coefficients. A sample of these surfaces is shown in 5a) a) b) Backscattering coefficient (dB) Figura 6: Comparison between HH and VV backscattreing coefficients of a gaussian surface when scalar Kirchhoff approximation (VV blue line with points and HH dashed red line with crosses) is applied with MoM solutions (continuous red and green lines) Kirchhoff MoM HH VV M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 8 Figure 6 compares HH and VV backscattering coefficients obtained by scalar kirchhoff approximation with the results obtained by MoM for the same gaussian profile. One sample of this profile has been shown in figure 4. It can be seen that they agree very well for incident angles closed to nadir direction. But the curves increase separation with the angle of incidence. This disagreement appears because we are not in the valid range of values of the product k·l for applying kirchhoff approximation, because Scalar Kirchhoff approach gives good results for: k·l>6, k· σ <2 and σ s<0.25. 5 CONCLUSIONS This paper demonstrates that fractal surfaces are a good tool for designing random rough surfaces. Here we have presented a very easy technique of generating band limited fractal structures. We have studied the variation of the roughness with the fractal parameters. On the other hand relations between fractal parameters and traditional statistical parameters have been presented. Further work is going to be held studying other fractal structures that can model more realistic surfaces. On the other hand, it is demonstrated that MoM is a suitable technique for obtaining the scattering coefficients of random surfaces. Its more important advantage, in front of analytical techniques, is that no hypothesis is considered. Consequently, sampling adequately the random surface, it can be applied to any kind of surface, with no restriction of roughness and at any range of frequency. The big drawback of MoM is that the computation time increases with the size compared to the wavelength and with the number of segments to be analyzed. Then it exists a compromise between accuracy and time consuming. The next step in the use of MoM is to apply a very optimized algorithm in a 2-D surface design embedded in a 3-D problem. ACKNOWLEDGEMENTS This work has been supported by the Spanish Comision Interministerial de Ciencia y Tecnología (CICYT TIC 99-1050-C03-01). REFERENCES [1] D.L. Jaggard and X. Sun, “Scattering from fractally corrugated surfaces”, Jour. Opt. Soc. Am., Vol. 7, No. 6 (June 1990). [2] R.M. Axline and M.F. Fung, “Numerical computation of scattering from a perfectly conducting random surface”, IEEE Trans. Antennas Propagation, Vol. AP-26, pp.482488 (1978). [3] M.F. Chen and S.Y. Bai, “Computer Simulation of Wave Scattering from a Dielectric Random Surface in Two Dimensions-Cylindrical Case”, Jour. Of Electromagnetic Wave M. Vall-llossera, N. Duffo, A. Camps, I. Corbella, J. Bará, F. Torres, M. Guillamont 9 and Applications, Vol. 4, No. 10, 963-982 (1990). [4] A.K. Fung and M.F. Chen, “Numerical simulation of scattering from simple and composite random surfaces”, Jour. Opt. Soc. Am., A/Vol. 2, No. 12 (December 1985).