1.Introduction

Space debris, which are unnecessary objects orbiting around the earth, can cause tragic accidents. In order to resolve this issue, Kamisaibara Space Guard Center (KSGC) radar system was built as the first radar facility devoted to space debris observation in Japan, and started the operation in 2004. Here we propose a fast algorithm of signal detection and orbit estimation for faint echoes from space debris.

2.The quasi-periodicity of the evaluation function
We approximate a space debris orbit as a straight line in the vicinity of the radar, and take the coordinate plane to include the orbit and the radar antenna. We can express the orbit with 3 parameters (r1, vd, s)=x, where (r1, vd, s) are the range, the Doppler velocity, and the angle between the direction of motion and that of the line-of-sight of the first echo respectively. We treat the orbit estimation process as an optimization problem, integrating received signals coherently and searching the optimum orbit parameters which maximize the output as

3.A fast orbit estimation algorithm using the quasi-periodicity
In order to resolve the problem of quasi-periodicity of the evaluation function, we propose a new orbit estimation algorithm.

We have experimentally clarified that the phase rotation p_i is dominant for the local shape of the evaluation function.

Therefore, in the local search, we fix other parameters to find the optimum p_i in (1). Once we find the local peak, we jump to the next area and update all the parameters.

We apply this algorithm to the experimental data. Fig. 2 shows an application example of the proposed method to real data for the H2A rocket booster, whose RCS is 27.6 m², which gives the peak SNR of 0.53dB higher than the conventional method. The error of the estimated parameters are substantially reduced to 140 m for r1 and 3.9 m/s for vd, respectively.

References
[1] K. Isoda, T. Sakamoto, T. Sato, IEICE Society Meeting No. B-2-12, Sep 2005.

Due to the international demand for renewable energy sources, many countries are realising the importance of wind power. As a consequence, a great quantity of wind-farms, consisting of increasingly large turbines, are being developed. Due to the large physical size of wind turbines, many objections are raised as the wind farm presents an electrically large obstacle at typical radar frequencies. In some countries, 50% of wind farm planning applications raise objections due to proximity to a radar installation. These objections are understandable, particularly for safety critical radar (Air Traffic Control, Defence) but objections are often raised without a complete understanding of the problem. In order to mitigate this problem, the interaction of wind turbines and radar systems must be understood. A final objective is to create a tool capable of quantifying the impact of proposed wind farms on radar installations.

A useful tool in understanding the effect of wind turbines on radar systems is computational modelling. A critical aspect of this is to understand the electromagnetic scattering behaviour of the wind-turbine in terms of Radar-Cross-Section (RCS) and Doppler signature. Typical wind-turbines have rotating aerodynamic blades which are 40-60m long, (with 100m blades being considered in the future) which produce a significant Doppler return. The tower is even larger and produces a large static return. There is also scattering from the nacelle, and interaction between each component. At the radar frequencies considered in this study, 3GHz and 10GHz, the electrical size of the turbines, together with features such as aerodynamic surfaces, present a challenge in electromagnetic modelling.

This paper reports the development of modelling techniques to predict the RCS and Doppler spectrum from wind-turbines. A review of suitable modelling techniques is presented, including both in-house and commercial tools. Detailed CAD models of the turbine structure, provided by wind-turbine manufacturers are used. The total RCS from the structure is decomposed to highlight the areas which contribute most significantly to the scattering cross-section. RCS and Doppler spectra will be presented for different orientation angles. Finally, recommendations for application of stealth technologies for different parts of the turbine will be discussed, with a prediction of efficacy.

This paper investigates the scattering behavior of large, ferroelectric inhomogeneous bodies with high permittivity, varying linearly along one dimension (x-axis). One of the salient features of such materials is their capability of changing their dielectric permittivity, and hence their operational characteristics, by applying a constant bias field. They are very useful in microwave, phased array and scattering applications, mostly in conjunction with E- and H- plane horn antennas. Mathematical modeling of their scattering properties is based on either Surface or Volume Integral Equation (SIE and VIE) formulations. In order to approximate the linear variation of the permittivity, a staircase scheme is utilized, and the dielectric slab is decomposed into a number of homogeneous, pencil-like sections (figure 1). The Radar Cross Section (RCS) and the induced currents are computed by increasing the number of sections until their values converge (figures 2 and 3). In general, RCS convergence is achieved faster. In the SIE case, the number of unknowns is generally low, but rises dramatically with the number of sections and the problem complexity may become computationally prohibitive. VIE with tetrahedral discretization is also invoked, provided that the total number of unknowns is tractable. The advantage of VIE over SIE is that the unknowns practically do not depend on the number of slab sections. For the cases where large systems arise (for high dielectric permittivity or high frequency of incident electromagnetic wave), fast methods are employed to overcome the computational bottleneck. Appropriate numerical quadrature integrations are employed to determine the currents and far-field parameters. Several lossy and lossless, linear and ferroelectric materials are investigated, most of them being used in applications involving microwave resonators.

Fig.1. Scattering geometry	Fig.2. RCS as a function of theta	Fig.3. Induced current

Acknowledgments:
The Project is co-funded by the European Social Fund (75%) and National Resources (25%) (Herakletus)

This contribution treats the microwave imaging of three-dimensional (3-D) inhomogeneous (lossy) dielectric objects embedded in a homogeneous background medium. The goal is to reconstruct the 3-D complex permittivity distribution within a given inversion domain from measurements of the scattered field which are taken in a number of points surrounding the domain for multiple incidences and possibly for multiple frequencies.

The non linear inverse scattering problem is solved in terms of the optimization of a cost function that involves the discrepancy between the measured and the calculated scattered field. We only optimize for the complex permittivity hence the fields in the inversion domain are eliminated by solving a forward problem in each iteration. Two versions of the well-known FFT-method for solving the volume integral equation are implemented. One is the weak form formulation of Zwamborn and van den Berg (IEEE transactions on microwave theory and techniques, 40(9), 1757-1766, 1992) and the other is a stronger form in which only the electric flux density is expanded in rooftop functions and where no averaging whatsoever is used. Moreover the latter code is able to use the Multilevel Fast Multipole Algorithm (MLFMA) to further speed up the computation and reduce the needed memory in some cases of large geometries. The effect of both methods on the reconstruction results is compared.

Newton-type optimization techniques, such as Gauss-Newton and quasi-Newton algorithms with line search, are applied to the inverse problem and compared. The standard least-square cost function is used and a regularization technique to alleviate the ill-posedness of the problem is investigated.

Reconstructions from simulated data are presented for some 3-D objects in the configuration of the bistatic microwave measurement setup of Institut Fresnel, France (J.M. Geffrin et. al., Inverse Problems, 21, S117-S130, 2005), where the transmitter and receiver can be moved on part of a spherical or a cylindrical surface surrounding the object. From measurements on 3-D objects, already carried out in this configuration, a realistic noise level is estimated and applied to the simulated data.

The Physical Optics (PO) method has been widely used when analysing electrically large EM problems. It considers only the illuminated parts of the geometry for computing the scattered fields, assuming null currents in the shadowed parts of the bodies under analysis. However, the shadowing problem can be even more problematic than the field calculation itself, specially when dealing with curved surfaces in complex geometries and multiple interactions, where many eclipse effects may appear. In these cases, illuminated regions must be obtained by using expensive minimization algorithms. The set of illuminated regions must be accurately determined, which implies many ray-tracing intersection tests. In this communication, an alternative treatment of the PO currents is considered. New electric and magnetic current expressions are proposed from the Equivalence Principle in order to bypass the shadowing problem. The currents calculated this way can be expressed in terms of current modes, providing an efficient storage and field calculation. A combination of these currents with some acceleration techniques, such as the angular Z-Buffer or quasi-analytical integration procedures, makes possible to maintain the PO efficiency when analysing simple cases, greatly improving it in more complex environments.

Because of the limited amount of information collectable form the scattering experiments and the intrinsic instability of standard inverse scattering problems, it is quite difficult and in several cases almost impossible to achieve reliable and satisfactory (in terms of resolution accuracy) reconstructions of the inaccessible scenario under test without recurring to an effective exploitation of the a-priori information or and iterate use of the scattering data. As a matter of fact, the necessary spatial resolution would require a significant and independent set of data non-available due to the band-limited nature of the scattered field although multi-frequency or multi-illumination/multi-view setups would used. Moreover, the large ratio between problem unknowns and independent field samples causes the presence of the local minima in the cost function defined in the arising optimization problem. In order to contemporarily address these drawbacks, recent developments reported in literature suggest splitting the original problem in a series of successive sub-problems according to the general strategy of 'divide and conquer'. Then, each partial solution is profitably used as initialization for solving the successive problem, which turns out to be more and more close to the original one even though characterized by a reduced complexity because of the information on its solution acquired at the previous steps. In such a framework, Massa et al. proposed in [1] a pixel-based approach (called Iterative Multi-Step Approach - IMSA) which improves at each step the resolution accuracy in a subset of the whole investigation domain by considering a more detailed multi-resolution description of the unknown scatterer profile.
In such a contribution, the integration of the multi-step strategy with a shaped-based representation of the scatterer is addressed. Unlike [2], at each step of the iterative process, the Regions-of-Interest (RoIs) defined at the previous step are processed through a level set method [3] in order to better estimate the shape and the homogeneous dielectric characteristics of the scatterer at the successive steps. According to the standard IMSA, different levels of resolution are used in the investigation domain (i.e., higher in the RoIs defined at the current step and with a decreasing order in those estimated at previous steps) and an enhanced reconstruction is carried out through a suitable optimization strategy [4].
References

[1] A. Massa et al, "A new methodology based on an iterative multiscaling for microwave imaging,'' IEEE Trans. Microwave Theory Tech., vol. 51, pp. 1162-1173, 2003.
[2] A. Massa et al., "Detection, location, and imaging of multiple scatterers by means of the iterative multiscaling method," IEEE Trans. Microwave Theory Tech., vol. 52, pp. 1217-1228, 2004.
[3] A. Litman et al., "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Problems, vol. 14, pp. 685­706, 1998.
[4] M. Donelli et al., "Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers," IEEE Trans. Microwave Theory Tech., vol. 53, pp. 1761-1776, 2005.

Proposed topic A22 Scattering, inverse scattering, microwave imaging
Contact Author: Prof. Rocco Pierri

The problem of localizing and reconstructing the shape of a buried object by means of electromagnetic waves is relevant in a wide class of applications, such as ground penetrating radar, remote sensing. In this framework, the tomographic approach is particularly interesting because it allows to achieve a spatial map of the geometric features of the region under investigation.

Here, we address the inverse scattering problem of recovering the position and the shape of unknown objects starting form the knowledge of scattered field once a known electromagnetic field impinges on the region under test. We consider objects buried in a homogeneous half-space, whose dielectric properties are known, separated by a planar interface and the scattered field is collected at the air-soil interface. We refer to "strong scatterers" representative of both perfect conductors and voids or cavities provided that in the latter case the soil permittivity is sufficiently larger than the free-space one.

The inversion algorithm searches for the unknown contour of the current density induced on the object surface. In particular, we move under the Kirchhoff or Physical Optics (PO) approximation which allows to simplify the problem. Accordingly, a linear inverse problem is faced and the Singular Value Decomposition (SVD) tool is exploited for analyzing and solving it.

It has been already pointed out how, in free space inverse scattering problems, this linear PO based inversion scheme can be fruitfully employed beyond the validity of the PO approximation, that is, in the resonance region and for interacting scatterers [1] and the inversion approach has been experimentally validated with measurements collected in controlled conditions [2].

In this paper we extend the solution approach for perfect conductors buried in the half-space geometry and deal with the assessment and the improvement of the performances of the inversion algorithm.

Next, we extend the approach to the reconstruction of the shape of voids embedded in an half-space by resorting to a distributional model similar to the one used for the perfect electric conductors. In particular the contour is searched for as the support of the surface magnetic equivalent current density modelled still according to the Kirchhoff approximation [3].

Reconstructions will be given for multiple conductors and voids in realistic measurement configurations. [1] R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri," Beyond Physical Optics SVD Shape Reconstruction of Metallic Cylinders", IEEE Trans. Antennas and Propagation , vol.. 54, pp. 655-665, 2006. [2] F. Soldovieri, A. Brancaccio, G. Leone, R. Pierri," Shape Reconstruction Of Perfectly Conducting Objects By Multiview Experimental Data", IEEE Trans. Geoscience and Remote Sensing, vol. 43, no. 1, pp. 65-71, Jan. 2005. [3] A. Liseno, R.Pierri," Imaging of voids by means of a physical-optics-based shape reconstruction algorithm", Journal of the Optical Society of America A, vol. 21, n. 6, pp. 968-974, Jun. 2004.

Topic: Radar cross section
Drazen S. Sumic, WIPL-D d.o.o., Bul. Mihajla Pupina 10E/1, 11070 Belgrade, Serbia and Montenegro, e-mail: drazen.sumic@wipl-d.com, phone: +381112124656
Branko M. Kolundzija, Dept. of EE, University of Belgrade, PO Box 35-54, 11120 Belgrade, Serbia and Montenegro, e-mail: kol@etf.bg.ac.yu, phone: +381113218329, fax: +381113015655
Contact person: Branko Kolundzija
Computation of RCS of electrically large bodies is memory and time-demanding. Various high frequency/asymptotic computational methods have been presented with the aim of reducing memory and time requirements, but with loss of accuracy.
The aim of this paper is to present an effective way of applying method of moments (MoM) to the simulation of electrically large scatterers. Used formulation is based on surface integral equations (SIE) and employs higher order basis functions (HOBFs). Therefore, very large scatterers are modeled with a relatively small number of unknowns, within the memory capacity of PCs. However, even with the reduction of unknowns enabled by HOBFs, solving large linear systems with a direct solver is a lengthy process. On the other hand a dense system matrix resulting from MoM applied to SIE+HOBFs is generally not suitable for iterative solution.

The proposed approach is to apply maximally orthogonalized HOBFs, combined with double-diagonal system preconditioning, in order to enable use of iterative solvers, as will be explained in the paper.

Analyzed fighter model was built using bilinear quadrilateral patches over which higher order polynomial expansions of currents were defined. This model, 12 meters in length, was simulated on up to 1.39 GHz, in which case its length was 55 wavelengths. The excitation was a plane wave coming at an elevation angle of 60 degrees. By subdividing patches whose side length was over a specified threshold, we controlled the maximum order of expansion on all patches in the model. Smaller subdivision threshold resulted in more patches, lower orders of expansion defined on them, and a larger number of unknowns. On the other hand, lower orders of expansion enabled faster iterative solver convergence. In each case, iterative solver was demanded to decrease the residuum to a level that guarantees good results agreement with the direct solver.

Fig 1 shows the duration of simulation for iterative solutions with different subdivision thresholds as well as direct solution with threshold set at 2 wavelengths. Detailed results will be presented in the paper. For example, it will be shown that at 1.39 GHz (29699 unknowns) iterative system solution needs 17 times less operations than the direct one. The total simulation time (including matrix fill-in and post-processing) in this case is 8.3 times shorter than in case of direct solver.

Radar cross section of the airplane is shown in Fig 2. It was calculated in 3601 directions in the incident plane at the frequency of 1.39 GHz, while the current distribution over the surface was calculated in 1,344,000 points. This required 29699 unknowns, occupying 8 GB of RAM, and was done in less than 4 hours on 1.4 GHz Opteron processor under Linux.
Fig. 1 Fig. 2

The problem examined here is the quantitative tomographic reconstruction of an unknown object from real data. We consider the case of free-space experiments where a set of ultra-wideband antennas, located along a horizontal line, successively illuminate a cylindrical object whose length is large compared to the largest wavelength used. The scattered field is measured along the same line under a multi-frequency, multi-incidence, multi-static, aspect-limited configuration.

Real data are collected in an anechoic chamber from the so-called SIMIS (Synthetic Impulse Microwave Imaging System) that uses an array of eight ultra-wideband Vivaldi-type ETSA antennas [1] associated with a vector network analyzer [2]. The total field, corresponding to the sum of the incident (in the absence of target) and scattered (contributed by the target) fields, is measured over a large frequency band (2-8 GHz). The scattered field is obtained either by a differential measurement (with/without the object) or by excluding the early time response (gating in the time-domain).

Reconstructions are performed using a bi-conjugate gradient nonlinear inversion algorithm [3] associated to an edge-preserving regularization technique [4]. The incident field of the Vivaldi antennas is numerically computed and incorporated into the reconstruction process [5].

References:

[1] E. Guillanton, J.-Y. Dauvignac, C. Pichot, and J. Cashman. A new design tapered slot antenna for ultra-wideband applications. Microwave and Optical Technology Letters, 19(4):286-289, November 1998.
[2] C. Pichot, J.-Y. Dauvignac, C. Dourthe, I. Aliferis, and E. Guillanton. Inversion algorithms and measurement systems for microwave tomography of buried objects. Proceedings of the 16th IEEE Instrumentation and Measurement Technology Conference, IMTC/99, volume 3, pages 1570-1575, May 24-26, 1999. Venice, Italy.
[3] C. Dourthe, C. Pichot, J.-Y. Dauvignac, L. Blanc-Feraud, and M. Barlaud. Regularized Bi-Conjugate Gradient Algorithm for Tomographic Reconstruction of Buried Objects. IEICE Transactions on Electronics, E83-C(12):1858-1863, December 2000.
[4] P. Lobel, L. Blanc-Feraud, C. Pichot, and M. Barlaud. A new regularization scheme for inverse scattering. Inverse Problems, 13(2):403-410, April 1997.
[5] I. Aliferis, C. Pichot, J.-Y. Dauvignac, and E. Guillanton. Tomographic Reconstruction of Buried Objects Using a Nonlinear and Regularized Inversion Method. In Abstracts of the International Symposium on Non-Linear Electromagnetic Systems (ISEM99), page 83, May 10-12, 1999. Pavie, Italy.

The detection of defects or cracks inside products is a key-problem in many industrial processes to be addressed by means of non-invasive (and of course non-destructive) inspection techniques. In such a framework, the effectiveness of tomographic approaches based on the use of interrogating microwaves has been demonstrated in inspecting dielectric or conductive materials. However, the solution of a NDT/NDE problem through inverse scattering techniques is still ill-conditioned and non-linear. On the other hand, it should be noticed that a lot of a-priori information on the scenario under test is available and such an information can be effectively employed for reducing the problem unknowns and to improve the detection/reconstruction results. In [1] Caorsi et al. proposed a global optimization technique based on the so called "Free Space Green Function" (FGA) aimed at detecting a single unknown defect inside a known homogeneous host medium. A suitable defined Genetic Algorithm (GA) procedure allowed the exploitation of the a-priori information for determining the geometric parameters of the defect (i.e., the position, the size and the orientation) and its electromagnetic properties.

A further improvement of the FGA-approach was achieved in [2] by introducing the numerical computation of the "Inhomogeneous Green's Function" (IGA) for limiting the region of interest to the area occupied by the crack thus enabling more accurate reconstructions and a significant reduction of the overall computational burden [3].
Notwithstanding the effectiveness of the IGA approach, its implementation considered the unrealistic situation of a single defect. In order to extend the method more realistic problems characterized by complex geometries and defects different both in shapes and sizes. Towards this end, two different approaches are taken into account. The former deals with a set of parallel GA sub-processes. Each GA considers a population of trial solutions coding a fixed number of cracks. The estimated solution is that with minimum fitness value among that defined by each sub-process. The other works with a single GA process. In such a case, a single pool of solutions describing different number of defects through multiple-length chromosomes is defined as well as a new set of genetic operators.
The results of a numerical assessment are shown and discussed in order to demonstrate the effectiveness of the proposed approaches both in terms of reconstruction accuracy and computational costs.

References

[1] S. Caorsi, A. Massa, and M. Pastorino, "A crack identification microwave procedure based on a genetic algorithm for nondestructive testing," IEEE Trans. Antennas Propagat., vol. 49, pp. 1812-1820, Dec. 2001.
[2] S. Caorsi, A. Massa, M. Pastorino, and M. Donelli, "Improved microwave imaging procedure for nondestructive evaluations of two-dimensional structures," IEEE Trans. Antennas Propagat., vol. 52, pp. 1386-1397, 2004.
[3] S. Caorsi, M. Benedetti, A. Massa, M. Pastorino, and A. Rosani, "Optimization approaches for the detection of subsurface defects," PIERS2004 in Pisa, Pisa, Italy, p. S6.01, Mar. 2004.