ELECTROMAGNETIC SCATTERING FROM NATURAL SURFACES: VALIDATION OF A FRACTAL APPROACH

In this paper we present a framework to validate recently developed electromagnetic scattering methods, which employ the fractal geometry to describe the surfaces. In particular, a fractal surface has been designed and built. Its characteristics have been tailored for X-band laboratory experiments and its building was performed by using cheap and light materials (cardboard and aluminum). It has been verified that the surface parameters were compliant with the required characteristics and that their values are in the range of validity of the Kirchhoff approximation theory for the evaluation of the electromagnetic scattered field. Then, the scattered field measurements have been performed in an anechoic chamber using a HP8510 network analyzer. The obtained data are currently being calibrated, in order to make them ready to be compared with the forecasted field. I. INTRODUCTION The introduction of the fractal geometry provided researchers involved in the remote sensing community a powerful tool for a reliable natural surfaces description (Mandelbrot (1)). In particular, the fractional Browninan motion (fBm) and the Weierstrass-Mandelbrot function (WM) are the most suitable functions to this aim, and their use in the evaluation of the scattering from natural surfaces allowed to obtain encouraging results (Franceschetti et al. (2)). Their use in conjunction with the Kirchhoff approximation (KA) and the small perturbation method (SPM) (Franceschetti (2) et al.), or the extended boundary condition method (EBCM) (Franceschetti (3) et al.) led to fit measured data better than the same scattering methods did with the classical Gaussian surface description. The main limits to a full comprehension of the obtained results are related to the lack of measured data. In fact, previous experiments did not relate the field scattered by natural surfaces to their fractal parameters, and the comparison between electromagnetic scattering data was inferred by means of best fit procedures. In this paper we present a measurement campaign planned to give an answer to this lack of data. In particular, in Section II, we define the fBm and the WM processes and we introduce the techniques to numerically design and put up the surface. In addition, the main steps of the surface realization as the superposition of a wrinkled aluminium foil on a cardboard long scale (from 0.5 cm up) topography are presented. Some comments on the analysis performed to verify that the surface characteristics are compliant with the given requirements are also provided. In Section III the expression of the backscattering coefficient in the frame of the KA is provided as a function of the fBm parameters. Then, it is verified that surface parameters are within the range of validity of the KA, and the corresponding electromagnetic scattered field is evaluated and plotted, in order to be compared with measured data. A brief description of the electromagnetic scattering measurements performed in anechoic chamber is also provided. A final discussion and the conclusions are reported in Section IV. II. SURFACE DESCRIPTION AND BUILDING It is widely known that the fBm is the most suitable model to account for the natural surfaces properties. It is a continuous everywhere but differentiable nowhere process defined in terms of the pdf of its increments. In particular, a surface z(x,y) is an fBm if for every x, y, x', y':