Measurement of the electromagnetic field backscattered by a fractal surface for the verification of electromagnetic scattering models

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 1777 Measurement of the Electromagnetic Field Backscattered by a Fractal Surface for the Verification of Electromagnetic Scattering Models Giuseppe Ruello, Member, IEEE, Pablo Blanco-Sánchez, Antonio Iodice, Senior Member, IEEE, Jordi J. Mallorquí, Member, IEEE, Daniele Riccio, Senior Member, IEEE, Antoni Broquetas, Member, IEEE, and Giorgio Franceschetti, Life Fellow, IEEE Abstract—Fractal geometry is widely accepted as an efficient theory for the characterization of natural surfaces; the opportunity of describing irregularity of natural surfaces in terms of few fractal parameters makes its use in direct and inverse electromagnetic (EM) scattering theories highly desirable. In this paper, we present an innovative procedure for manufacturing fractal surfaces and for measuring their scattering properties. A cardboard–aluminum fractal surface was built as a representation of a Weiestrass–Mandelbrot fractal process; the EM field scattered from it was measured in an anechoic chamber. A monostatic radarlike configuration was employed. Measurement results were compared to Kirchhoff approximation and small perturbation method closed-form results that were analytically obtained by employing the fractional Brownian motion to model the surface shape. Matching and discrepancies between theories and measurements are then discussed. Finally, fractal and classical surface models are compared as far as their use in the EM scattering is concerned. Index Terms—Electromagnetic scattering by rough surfaces, fractals. I. INTRODUCTION ARELIABLE theory on the electromagnetic (EM) scattering from natural surfaces requires an efficient and accurate quantitative description of the natural surface shapes. The use of fractal geometry for this purpose is strongly suggested because it accounts for the complex behavior of nature by means of simple models. Elegant methods for the evaluation of the EM field scattered from fractal surfaces were recently developed with encouraging results [1], [2]. So far, no scattering measurement campaigns on surfaces with known fractal parameters were carried out. Most of the measurement campaigns presented in the literature Manuscript received September 5, 2007; revised June 19, 2008, December 12, 2008, and July 3, 2009. First published December 22, 2009; current version published March 24, 2010. This work was supported by the Spanish MCYT and EU FEDER funds under Project TEC2008-06764-C02-01. G. Ruello, A. Iodice, D. Riccio, and G. Franceschetti are with the Department of Electronic and Telecommunication Engineering, University of Naples “Federico II,” 80125 Naples, Italy (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). P. Blanco-Sánchez, J. J. Mallorquí, and A. Broquetas are with the Department of Signal Theory and Communications, School of Telecommunication Engineering, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2009.2036007 were performed on artificially constructed Gaussian surfaces [3]–[5] or on natural surfaces described by means of nonfractal parameters like the correlation length and standard deviation [1]. However, already in the 1970s, it was known that classical Gaussian parameters estimated from samples of natural surfaces depend on the size of the considered surfaces [6], [7]; this is considered as a major motivation for conceiving more appropriate models to describe natural surfaces. In 1973, Beckmann [6] expressed the need of a non-Gaussian description of natural surfaces for electromagnetic scattering evaluation purposes. He pointed out that the probability density function (pdf) of a natural surface can also be described by a Gaussian model, but the pdf could be noncritical for its scattering properties. In 1988, Wu et al. [8] agreed that non-Gaussian processes would be necessary for natural surface description. The introduction of the fractal geometry provided a powerful instrument for comprehending and quantitatively describing the irregular shapes of nature. As a matter of fact, fractal models account for the nonstationary and self-affine characteristics of actual processes, so that in the last decades, several researchers intensively used a fractal representation of natural surfaces (e.g., [1], [9]–[17] are a small but significant part of the huge literature on the topic). The use of fractals in remote sensing is highly desirable, also because the natural surfaces can be described by employing just few parameters, and this is very attractive for model-inversion purposes [18], [19]. In this paper, we present an innovative measurement campaign in a controlled environment carried out to test, in monostatic configuration, the theories on the electromagnetic scattering from fractal surfaces. In Section II, we introduce the fundamentals of fractal models, and we recall the main concepts which inspired the surface construction. In particular, we define the fractional Brownian motion (fBm), an everywhere continuous but nowhere differentiable process, which is the most suitable model to describe natural surfaces, and it is currently used in EM scattering theories. Then, in order to synthesize an fBm fractal surface, we introduce the Weierstrass–Mandelbrot function (WM) whose spectrum provides a reliable approximation of an fBm process spectrum. The values of the fractal parameters chosen for the numerical synthesis of the surface are consistent with typical natural surfaces. Then, we present the procedure to build the surface as a superposition of a cardboard structure, representing the low frequencies, and wrinkled aluminum foils, accounting for the high-frequency surface components. Employed 0196-2892/$26.00 © 2009 IEEE Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. 1778 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 materials guaranteed to produce a cost-effective, portable, and reliable fractal surface. The correspondence between synthesized and built-surface fractal parameters was checked by means of appropriate optical measurements [20]. In Section III, we recall how the fBm surface description can be used in conjunction with the Kirchhoff approximation (KA) and the small perturbation method (SPM) [1], in order to provide the second-order statistics of the scattered field [21], [22]. The choice of the KA and the SPM is related to the fact that these are the most used techniques in remote-sensing applications: from SAR data interpretation [23] to retrieving algorithms [18], [19]. Section IV is devoted to describe the procedure and the implementation of the EM scattering measurements, performed in the anechoic chamber of the Universitat Politècnica de Catalunya (UPC). A couple of X-band horn antennas was used to illuminate the built surface and measure the field backscattered from it. The built surface was mounted on a rotor, whose movements allowed the acquiring of a sufficient number of samples at different incidence angles (incidence angles in the range of 0◦−70◦were explored). Both horizontaland verticalpolarized scattered fields were measured. The calibration procedure performed with the use of a trihedral corner reflector is also presented. In Section V, we show the obtained results. Calibrated measured data are presented as a function of the incidence angle; they are then compared with the theoretical results in Section III. Matching and differences are deeply discussed. In addition, a discussion on the main advantages of fractals with respect to classical models in the EM scattering methods is presented. The aim of Section VI is to summarize the rationale, approach, implementation, and results related with the presented work. II. SURFACE CONSTRUCTION In this section, we recall the basic principles of the fractal geometry, and we provide the rationale for the synthesis, the building, and the validation of the fractal surface employed in our experiments. A detailed description of these topics is provided in [20]. A. Fractal Geometry The fBm is widely recognized as the most suitable fractal process to model natural surfaces. It allows the describing of the surface shape in terms of only two parameters, the dimensionless Hurst coefficient,H, and the incremental standard deviation at unitary distance,s, measured in m(1−H).AnfBm process represents a stochastic surface z(x, y)if, for every x,y, x, and y, the pdf of its increments is Gaussian with zero mean and standard deviation sτH Pr z(x, y)−z(x,y)<¯ ζ =1 √2πsτH ¯ ζ  −∞ exp −ζ2 2s2τ2Hdζ (1) where τis the distance between x,yand x,y. It can be demonstrated [13], [24] that a process satisfying (1) exists if 0<H<1, and that (with probability one) any fBm sample surface has a fractal dimension D=3−H. Furthermore, if we consider the increments at distance τ=1m, their standard deviation is s. Therefore, the sparameter represents an index of roughness. The higher its value, the rougher the surface [1]. Evaluation of the fBm power spectrum deserves special care due to the nonstationarity of the surface. Space-frequency and scale-frequency approaches are required in order to express the power spectral density of the surface. Both these approaches lead to a power-law spectrum [1], [13] W(k)=S0k−α(2) where S0and αare parameters depending on Hand s[1] and k is the spatial wavenumber. The fBm is a regular stochastic process, whose synthesis is not a straightforward task, and it is usually accomplished in terms of appropriate functions, as detailed in the following section. B. Synthesis The synthesis of fBm processes can be obtained via displaced interpolation [25], spectral synthesis [1], wavelet [25] methods, and so on. In this paper, we use a WM function, which is a predictable process that allows an easy and controllable modeling of deterministic and stochastic processes as a function of few physical parameters. A mathematical WM is a superposition of infinite sinusoidal tones with periods spaced by an irrational factor. Natural surfaces exhibit a fractal behavior in a wide but finite range of scales, so a physical WM [1] can be obtained by using a 2-D band-limited (i.e., with a finite number of tones, M)WM function f(x, y) f(x, y)=B M−1  n=0 Cnν−Hn sin(k0νn(xcos ψn+ysin ψn)+φn) (3) where Bis an amplitude scaling factor; νis the irrational frequency scaling factor; Cn,ψn, and φnare random variables, accounting for amplitude, direction, and phase behavior of each tone, respectively [1]; and k0is the fundamental tone wavenumber. It was shown that the WM can be considered as a spectral sampled version of an fBm process with the same fractal dimension and with S0related to the WM parameters by [1] B2=S0 2πH k−2H 0(νH−ν−H).(4) The synthesis of a single surface calls for the setting of five parameters: H,B,k0,kM−1, and ν. For the surface considered in this paper, their values were chosen according to the following rationale. 1) The Hvalue was set at 0.7, because natural surfaces typically hold Hvalues ranging from 0.6 to 0.9 [24]. 2) The Bvalue was set at 0.011 m, in accordance to typical values for natural surfaces. This choice corresponds to an svalue of 0.063 m1−H, in accordance with typical natural values [24]. Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. RUELLO et al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1779 TABLE I SIMULATED SURFACE PARAMETERS 3) The lowest roughness scale k0that contributes to the scattered field formation depends on the illuminated area and the anechoic chamber dimensions. In this paper, the fundamental tone wavenumber is of 5.71 m−1. 4) The highest roughness scale kM−1that contributes to the scattered field formation depends on the incident wavelength λ. Scales lower than a fraction of the wavelength do not contribute to the field formation. We fixed λ= 3cm as a reference wavelength, so that reliable measurements could be performed at frequencies included in the X-band. Therefore, kM−1= 1943.8m−1, that corresponds to a surface tone wavelength of 3.2 mm, which is about λ/10. 5) The νvaluewassetat0.5e, as a tradeoff between the fBm sampling rate and the memory constraints. Such a choice leads to a number of tones of M=20[20]. In Table I, the chosen numerical values are summarized, and a representation of the synthesized surface is shown in Fig. 1(a). C. Manufacturing The surface was built according to a two-step approach. The largest spatial scale roughness (from meters till 0.5 cm) was assembled as superposition of cardboard layers shaped according to the synthesized surface-level curves with a step of 0.5 cm (i.e., λ/6) [see Fig. 1(b)]. The choice of cardboard allowed an easy manufacturing and portability of the surface. Once such a 1.5 m ×1.5 m (i.e., 50 λ×50 λ) macroscopic structure was built, we added the microscopic roughness by superposing two layers of aluminum foils [see Fig. 1(c)]. The first layer was glued directly to the cardboard without corrugation. The second was manually wrinkled and glued on the top. The intensity of the corrugations was not precontrolled but evaluated via laser scansion. Such a procedure allowed the considering of the built surface as perfect reflector. In Fig. 1(d), we show a top view of the built surface. Note that the aluminum foils cover up the seams, avoiding the possibility that they could bias the results with their nonfractal straight edges. The chosen surface shape is circular, with a 1.5-m diameter, in order to minimize the border effects [20]. D. Validation A high-precision laser (with resolution of 0.7 mm) was employed to provide an accurate analysis of the built-surface properties. The measured data were processed in both the spatial and spectral domains, leading to the result that the fractal surface holds the prescribed synthesized parameters [20] within the range of scales of interest for electromagnetic scattering experiments [1]. The obtained measurements showed that the aluminum layers did not significantly change the surface roughness at the scales of interest for the electromagnetic scattering. Fig. 1. (a) Synthesized surface. (b) Cardboard topography. (c) Detail on the aluminum layers. (d) Top view of the built surface. III. ELECTROMAGNETIC METHODS In this paper, we focus our attention on methods that provide simple and effective closed-form solutions, because their use is of interest in remote sensing applications [1], [18], [19], [26]. In particular, we focus our analysis on the mostly used electromagnetic methods, the KA and the SPM. These methods are intensively used in remote sensing applications for developing Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. 1780 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 SAR interpretation tools [23] as well as inverse methods for retrieving physical parameters in remote sensing applications [18], [19]. The introduction of the fBm fractal process for surface profile description purposes allowed the development of improved versions of these electromagnetic scattering methods [1], [2], [21], [22], whose results are recalled in the following. A. KA Formulation The PO solution of the KA leads to express the scattered power density Sias a function of the surface parameters [1], [2] Si∝ ∞  0 J0(ηxyτ)exp−1 2η2 zs2τ2Hτdτ (5) where J0is the zeroth-order Bessel function and η=ki−ks=(ηx,η y,η z), where kiand ksare the incident and scattered wave vectors, respectively, ηxy =η2 x+η2 y.The surface roughness, expressed in terms of the fractal parameters sand H, as well as the incidence angle are accounted for in the arguments of the Bessel and the exponential functions. A complete treatise of the integral evaluation is well beyond the goals of this paper. In literature, closed-form solutions can be obtained via asymptotic expansions, as detailed in [1], [2]. B. SPM Formulation The SPM formulation provides the radar cross section as a function of the surface spectrum. If we use the fBm spectrum in the expression of the first-order SPM radar cross section, we obtain the radar cross section as a function of the spectral surface parameters [1], [2], [22] σo pp =4k4cos4ϑ|βpp|2S0 π(2ksin ϑ)α(6) with βpp taking into account the polarization issue, and S0and αare the fBm power-law spectrum parameters [1], [2]. C. Validity Limits The definition of the validity limits for electromagnetic scattering from fractal surfaces is still an open problem. This is mainly due to the fact that the fBm is not differentiable, and its use in the scattering evaluation requires a physical-based bandlimiting procedure. Such an operation influences the definition of the validity limits. So far, the validity limits are evaluated via relationships between fractal and classical parameters, as reported in [1], [2]. However, the equivalence between classical and fractal parameters requires subtle theoretical issues. To the best of our knowledge, no conclusive results exist in the available literature. In a recent paper [26], an empirical condition was defined in terms of the significant slope ss(the ratio between the root mean square (RMS) surface height and the wavelength of the dominant spectral peak). According to [26], for near-nadir incidence (incidence angles lower than 30◦), the PO approach is applicable if ss<0.037 cos3θ.Forthe surface employed in our experiment, the condition holds for every incidence angle less than 30◦. Despite the fact that it does Fig. 2. (a) Top view of the anechoic chamber. A identifies the antenna position, B identifies the surface position. Linear dimensions are expressed in centimeters. (b) Image of the transmitting and receiving antennas. not constitute a conclusive proof, this result suggests that, at least for near-nadir angles, validity limits are nearly fulfilled. A complete discussion on this topic requires a reformulation of the validity limits in terms of fractal parameters, but it goes beyond the goals of this paper and it is demanded to a future discussion. IV. ELECTROMAGNETIC MEASUREMENT SETUP A. Geometry The fractal surface, built in accordance with the procedure of Section II, was used to validate the EM scattering methods presented in Section III. In this section, we present the procedure to measure the EM field backscattered from this surface. The experiments were performed in the anechoic chamber at UPC. A top view of the measurement geometry is shown in Fig. 2(a). Two horn antennas, one for transmitting and one for receiving [see Fig. 2(b)], were placed in a fixed position [see the A point in Fig. 2(a)]. The surface was mounted in position B [see Fig. 2(a)] at a distance of 5.75 m from the antennas, on a rollover azimuth positioning system that allows rotations both in Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. RUELLO et al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1781 Fig. 3. Measurement geometry. The rotor allows the built-surface rotation around the (a) z-and(b)x-axes. the (y,z)plane around the x-axis and in the (x, y)plane along the z-axis, in accordance with the geometry of Fig. 3. The spot size of the illuminating beam covers the whole surface. The rotations around the z-axis allowed the acquisition of scattered field samples at different incidence angles θ[see Figs. 2 and 3(a)]. At a given incidence angle, the rotations around the x-axis guarantee the acquisition of many independent scattered field samples. B. Measurement Procedure For each surface (θ,φ)position, the X-band antennas acquired 401 samples, uniformly spaced within the frequency range from 7 to 12 GHz. Appropriate transmitting and receiving antenna rotations provided the acquisition of HHand VVpolarized fields. The field backscattered from the fractal surface was measured using an HP8510 network analyzer for θranging from 0◦to 70◦witha2 ◦step. For each θvalue, 72 independent field values were acquired by rotating the surface with φsteps of 5◦. In fact, the fractal surface under analysis is a single realization of an fBm stochastic process. It can be seen as a cell composed of multiple scatterers. By changing the illumination angle, the relative distribution of the scatterers changes, so that two acquisitions are different. The rotation of a resolution element by the angle φmoves the scattering centers. The range component of the displacement causes a slightly different phase shift for each scattering center, resulting in signal decorrelation. We choose aφstep of 5◦, in accordance with the results found in the literature concerning the signal decorrelation for interferometric applications [27], [28]. In [27], the correlation coefficient Fig. 4. Comparison between (dots) data, (dashed line) KA, and (solid line) SPM for (a) VV and (b) HH polarizations. between the field scattered by a surface and the field scattered by the same surface rotated by an angle φis computed as ρφ=1−2(sin θ)|φ|r λ,if 2(sin θ)|φ|r λ<1 0,otherwise (7) where ris the antenna–surface distance. In fact, by using in this expression φ=5 ◦, considering the values of λand θbelonging to the intervals defined earlier and considering r=5.75 m (see Fig. 2), we obtain a null value of the correlation coefficient. C. Calibration The calibration is performed via a standard two-step procedure [29], [30]. First, the power Pmbackscattered by the built surface, placed at a distance rmfrom the antenna, is measured to obtain the non-calibrated data Pm=PtG 4πr2 m σm 4πr2 m K. (8) It depends on the antenna gain G, the transmitted power Pt, and the environmental effect K. Then, the surface is replaced by a perfectly conducting trihedral corner reflector, and the received power Ptri is measured at a distance rtri Ptri =PtG 4πr2 tri σtrri 4πr2 tri K. (9) Therefore, the calibrated radar cross section σmis obtained by the known trihedral radar cross section σtri as Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. 1782 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 Fig. 5. VV polarization. Comparison between (dots) data, (dashed line) KA, and (solid line) SPM as a function of the frequency for incidence angle of (a) 15◦, (b) 30◦,and(c)45 ◦. follows: σm=r4 m r4 tri Pm Ptri σtri.(10) Based on previous measurement campaigns, the measurement experimental error is expected to be lower than 1 dB [31]–[33]. V. R ESULTS In this section, we present a comparison between the data measured in accordance with the procedure described in Section IV and the results obtained by employing the theoretical methods recalled in Section III. The set of acquired scattering data allows the investigation of the properties of the backscattered electromagnetic field as a function of the incidence angle and of the electromagnetic frequency. A. Overall Comparison In Fig. 4(a), a comparison between measured data (dots), SPM (solid line), and KA (dashed line) is shown for VV polarization. As stated in Section III, both theoretical methods make use of the fBm process for the surface description. Each dot is representative of the value averaged over all the φangles and the frequency-band measurements. The theoretical curves are also averaged in the considered band. Note that at low incidence angles (up to about 20◦), KA predictions well match Fig. 6. HH polarization. Comparison between (dots) data, (dashed line) KA, and (solid line) SPM as a function of the frequency for incidence angle of (a) 15◦, (b) 30◦,and(c)45 ◦. TABLE II VARIANCE AND CORRELATION LENGTH VALUES AS A FUNCTION OF THE OBSERVED AREA the experimental data; at larger incidences, we expect that KA validity limits are not satisfied, and the electromagnetic method accuracy decreases. The SPM seems to be able to follow the field behavior for intermediate incidence angles, in accordance with its validity limits [34]. Similar conclusions can be inferred by analysis of Fig. 4(b) for HH polarization. Anyway, a complete discussion on the method validity calls for a reformulation of limits in terms of fractal parameters. B. Frequency Dependence The field scattered by a surface is strongly dependent on the EM incident wavelength; hence, an accurate study on the dependence of the measured data on the frequency is required in order to verify that the performed averages make sense. Therefore, we investigated the EM scattered field of Fig. 4 as a function of the field frequency for fixed incidence angles, (see Figs. 5 and 6, respectively) for VV and HH polarizations. Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. RUELLO et al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1783 Fig. 7. VV polarization. Comparison between (dots) experimental results and the KA results in conjunction with (solid line) fBm, (long dashed line) Ga-Ga, and (short dashed line) Ga-Exp for as follows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. In Figs. 5(a) and 6(a), we show a comparison between measured data (dots), SPM (solid line), and KA (dashed line) as a function of the field frequency, for θfixed at 15◦.Atthis incidence angle, there is a good matching between measurement data and prediction provided by both theoretical models, see Fig. 4. Apart from small random oscillations, measured data show a frequency behavior similar to those of the theoretical curves. Such a result confirms that averaging data over the employed frequency range is meaningful for comparative purposes. In Figs. 5(b) and 6(b), the θvalue is fixed at 30◦. We still note that the frequency distribution of the acquired field values is almost linear with a law similar to that of the theoretical method results, again justifying averaging data over the employed frequency range for comparison purposes. For the highest frequencies, we can note a worst agreement between the theoretical models predictions and the experimental data results, due to the fact that the built-surface roughness is scarcely controlled at the smallest spatial scales, which are involved in the scattering at the highest frequencies. In addition, at the highest frequencies, the electromagnetic methods are not completely adequate to model the observed phenomenon, because they do not account for shadow and multiple-reflections phenomena. The reduced frequency band where this behavior occurs does not impair the comparison results. In Figs. 5(c) and 6(c), the θvalue is fixed at 45◦, in correspondence of a significant mean difference between data and experiments, see Fig. 4. Again, the frequency distribution of the acquired field values is almost linear with a law similar to that of the theoretical method results, justifying averaging data over the employed frequency range for comparison purposes. So far, we limited our attention on SPM and KA based on the fBm surface description. Despite it is widely accepted that natural surfaces are efficiently described by fractal models, we find it useful to compare fractal and classical methods. The results of such a comparison can drive the choice of the surface model to be employed in remote sensing applications. C. Gaussian Versus Fractal Surface Models In the following, we compare the experimental data with the results of the KA and SPM electromagnetic methods obtained by employing fBm and classical surface models, i.e., a Gaussian surface pdf, with Gaussian (Ga-Ga) or exponential (Ga-Exp) correlation function. In order to perform the comparison, it is necessary to determine the correlation length and the surface height variance. Therefore, we estimated these parameters on the largest possible square area (90 ×90 cm2)of the built surface: The obtained results are presented in the first row of Table II. In Figs. 7(a) and 8(a), the comparison between the measured scattered fields (dots) and the theoretical results obtained by using fBm (solid line), Ga-Ga (long dashed line) and Ga-Exp (short dashed line) surface models in the KA is provided for VV and HH polarizations, respectively. In Figs. 9(a) and 10(a), the same analysis is shown for the SPM method. It is evident that the fBm surface model better matches the data with respect to classical models, at least for small and moderate incidence angles. The only exception regards the HH polarization for the SPM case. The differences between fractal and classical scattering model results are surprisingly wide, and they lead to think that the used classical parameter values are not representative of the considered surface. Therefore, we evaluated the classical parameter values land σby considering smaller portions of the built surface (see from the second to the fourth rows of Table II), and we compared the correspondent evaluated EM scattered field with the measured data, as shown in Figs. 7(b)–(d), 8(b)–(d), 9(b)–(d), and Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. 1784 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 Fig. 8. HH polarization. Comparison between (dots) experimental results and the KA results in conjunction with (solid line) fBm, (long dashed line) Ga-Ga, and (short dashed line) Ga-Exp for as follows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. Fig. 9. VV polarization. Comparison between (dots) experimental results and the SPM results in conjunction with (solid line) fBm, (long dashed line) Ga-Ga, and (short dashed line) Ga-Exp for as follows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. 10(b)–(d). As expected, for classical surface descriptions, land σvalues depend on the dimension of the surface used to evaluate them. Figs. 7–10 show that the variations of the land σvalues strongly affect the evaluation of the scattered field. As a dramatic consequence, this strongly reduces reliability of results obtained by employing classical scattering theories and their use for inverse problems in remote sensing applications. Note that we can obtain a reasonable agreement between measurements and data by appropriately choosing the Gaussian parameters σand l: for instance, in the considered case study, the KA with Ga-Ga surface model with σ=1.3cm and l=9.1cm allows the obtaining of a rather good agreement Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply. RUELLO et al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1785 Fig. 10. HH polarization. Comparison between (dots) experimental results and the SPM results in conjunction with (solid line) fBm, (long dashed line)Ga-Ga, and (short dashed line)Ga-Exp for as follows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. with measurements for angles lower than 20◦, [see Figs. 7(b) and 8(b)]. However, the σand lvalues that guarantee this agreement are not representative of the surface. The obtained results lead to the conclusion that fractal models are more appropriate than classical ones, for the following reasons: 1) the electromagnetic results are closer to experimental data, and 2) the fractal parameters better describe the surfaces because they are intrinsic parameters, i.e., their evaluation does not depend on the observer. Both results suggest the use of fractals for the electromagnetic scattering evaluation as well as for inverse remote-sensing problems. VI. CONCLUSION In this paper, we presented an innovative measurement procedure for validating the theoretical methods of evaluation of the EM field scattered from natural surfaces. To this aim, a fractal surface with assigned parameters was built as a superposition of cardboard and aluminum layers, as presented in a recent paper. The characteristics of such a surface were verified by measuring it with an optical high-precision instrument. The surface was mounted on a rotor in an anechoic chamber, and the EM field backscattered from it was measured at different incidence angles. Problems related with the acquisition of the field values were addressed and solved. The comparison between the obtained calibrated data and the theoretical results deriving from the fBm use in the KA and SPM methods shows matching and discrepancies between theoretical prediction and experimental results. In particular, at low incidence angles, the KA appears to be the most appropriate method to predict the scattered field, whereas at intermediate angles, the SPM better matches the data, as predicted by theory. In addition, provided that the natural surfaces are well described by fractal laws, we explored the possibility that classical methods relying on the Gaussian surface description could be used for the evaluation of the field scattered from natural surfaces. Therefore, we measured the standard deviation σand the correlation length lon surface portions with different sizes, and we compared the predicted EM scattered field with data. The obtained σand lvalues and, as a consequence, the forecasted scattered field values turn out to depend on the chosen dimension. It means that the classical σand lparameters are not intrinsic descriptors of the surface. The obtained experimental results lead to the conclusion that the use of fractal surfaces in the EM scattering theories provides two main advantages: 1) the results are closer to experimental data, and 2) the surface is efficiently described in terms of only two intrinsic parameters, whose value does not depend on the observer. REFERENCES [1] G. Franceschetti and D. Riccio, Scattering, Natural Surfaces, and Fractals. Burlington, MA: Academic, 2007. [2] G. Franceschetti, A. Iodice, and D. Riccio, “Fractal models for scattering from natural surfaces,” in Scattering, R. Pike and P. Sabatier, Eds. 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Remote Sens., vol. 38, no. 2, pp. 630–640, Mar. 2000. Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 at 14:29:17 UTC from IEEE Xplore. Restrictions apply.