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Many natural and man-made rough surfaces exhibit multiscale features and may be advantageously described by fractal models. The surface roughness is then characterized by the fractal dimension while the distribution of gap or scale sizes (or gapiness) is measured by the lacunarity. Fractal dimension and lacunarity are two independent descriptors, that is, two surfaces having the same fractal dimension may look different and differ according to the distribution of gap or scale sizes. The remote analysis of the scaling properties of such multiscale surfaces may be achieved from interrogation by an electromagnetic pulse. Numerous approaches have been developed to describe the relationship between the fractal dimension of rough surfaces and the properties of the scattered waves. But, despite its potential importance in engineering, the remote extraction of the lacunarity from scattering data has not received the same attention. This paper is devoted to the extraction of the lacunarity of rough (fractal) surface from the wavelet analysis of the near-field impulse response using a heuristic method. It is shown that the skeleton of the wavelet-transform modulus-maxima of the scattered signal allows the straightforward estimation of the surface lacunarity for the examples studied here. Finally the wavelet analyze is applied for studying rough surfaces of different lacunarities and various fractal dimensions. Index Terms--Inverse problem, lacunarity, multiscale, rough surfaces, wavelet analysis.

A method to build impedance boundary conditions incorporating the effect of rapid variations of a perfectly conducting surface on the scattering of a scalar, E-polarized, time-harmonic electromagnetic wave is presented. The amplitude and the extent of the variations are assumed to be comparable to each other and small as compared to the wavelength. The derivation of the impedance boundary conditions is based on a decomposition of the field in two parts. The first part describes the overall behavior of the wave and the second one deals with its small scale variations. The effective boundary conditions are rigorously constructed for periodic surfaces presenting a large-scale global periodicity to suppress the boundary effects and a small local period to describe the rapid variations. Numerical examples prove that the method can even be heuristically extended to more general problems. In this respect, there are reported some results related to the numerical treatment of small details on a smooth surface and of rough surfaces without resorting to refined meshes. Index Terms--Electromagnetic scattering, homogenization, impedance boundary conditions, periodic surfaces, rough surfaces, small details.

In this paper we present an efficient technique for solving dense complex-symmetric linear system of equations arising in the method of moments (MoM) formulation. To illustrate the application of the method, we consider a finite array of scatterers, which gives rise to a large number of unknowns. The solution procedure utilizes preconditioned transpose-free QMR (PTFQMR) iterations and computes the matrix-vector products by employing a compressed impedance matrix. The compression is achieved by reduced-rank representation of the off-diagonal blocks, based on a partial-QR decomposition, which is followed by an iterative refinement. Both the preconditioning and the compression steps take advantage of the block structure of the matrix. The convergence of the iterative procedure is investigated and the performance of the proposed algorithm is compared to that achieved by other schemes. The effectiveness of the preconditioner and the degree of matrix compression are quantified. Finite arrays of variable shape and sizes are considered, and it is demonstrated that the ability to solve large problems using this technique enables one to evaluate the edge effects in the finite array. Such array is basically flat and periodic, but the algorithm is still efficient when variation with strict periodicity or flatness exists. Index Terms - Electromagnetic scattering, moment methods, numerical analysis.

We present here a multilevel version of a compression technique for method of moments (formerly presented). This method is based on QR compression of the impedance matrix off-diagonal blocks. The block structure of the matrix is built up to optimize the compression rate. It results from the study that compression threshold is the unique adjustment parameter, which makes this method very easy to handle. Some results are presented, which stress the improvement in memory storage and computation time obtained through this approach with respect to the mono-level algorithm. Index Terms--Compression methods, integral equations, method of moments (MOM), QR decomposition.

A new approach based on QR compression of the off diagonal blocks of the impedance matrix arising in integral equations solutions of electromagnetic (EM) scattering problems, previously implemented for finite and irregular gratings, is extended here to an arbitrarily shaped obstacle. The main tool consists in dividing the entire surface scatterer into subdomains and in assembling the global impedance matrix as a combination of the macromatrices associated to each interaction of any patch on the other. Correlation between compression rate and accuracy of field computation is drawn up. An experimental study is performed to optimize the method relatively to the two competing parameters: accuracy and cost. Finally, it is shown that this method works equally well in both the high and the low frequency domain and also with an inclusion of a small detail in a large object. Index Terms--Electromagnetic (EM) scattering, large size problems, matrix compression, method of moments (MoM).