DISEÑO DE SOFTWARE PARA EL DISEÑO DE UN INDUCTÓMETRO

The aim of this study w as to investigate w ave scattering from perfectly- conducting, tw o-dim ensional, G aussian ro u g h surfaces w h ere the RMS height an d correlation-length of the surface are of the sam e order, and of the sam e o rd er as the electrom agnetic w av elen g th . The in v estig atio n started w ith the tw o, uncoupled, m agnetic-field-integral-equations (MFIEs), an d the p ro ced u re u sed to ap p ro x im ate the co n tin u o u s e q u atio n as a discrete eq u atio n . For the surfaces w e have co n sid ered , the m atrices g en erated in the d iscretizatio n of the co n tin u o u s eq u atio n are n o t ill- conditioned and can be solved exactly by LU decom position. We chose to investigate the quality of the num erical solution by exam ining the degree to w hich the scattered field b en eath the surface b o u n d ary cancelled the incident field. This established th at the discrete approxim ation of the MFIE was a good one.

Once confident th at the pro ced u re u sed to discretize the MFIE gave good solutions to the field in the vicinity of the surface boundary, the study p rogressed onto iterative m ethods of solving the discrete equation. The convergence a n d rate of convergence of tw o iterativ e m eth o d s w ere exam ined. The N eum ann expansion used by Brown (1982), H olliday (1987),

and H olliday et al (1988), appeared to be a natural candidate for an iterative solution of the discrete approxim ation of the MFIE. H ow ever, although the expansion pro v id ed a rap id num erical solution for small values of surface height an d slope, w hen the surface structure w as of the sam e order as the electrom agnetic w avelength the expansion diverged rapidly. A step-by-step m ethod of identifying divergence w as p resen ted (W ingham an d Devayya, 1992). This allow ed u s to id en tify d iv erg e n t expansions w ith in a few iterations. To the extent th at the num erical sim ulation is a good one, w e also consider th at o u r results provide strong evidence th at the N eu m an n ex p an sio n can n o t be u sed w ith o u t q u alificatio n to p ro v id e a form al solution to the ro u g h surface MFIE.

The co n ju g ate-g rad ien t m eth o d s are iterativ e m eth o d s of solving m atrix-equations w hose convergence are in th eo ry sure. In spite of the theoretical assurance of convergence, it is n o t uncom m on to find in the literature references to the iteratio n diverging. We have ourselves found th at applied to the discrete approxim ation of the MFIE, convergence is not sure. The divergence w as identified as due to the effect of ro u n d in g errors on the theoretical orthogonality properties, w hich guarantee convergence. To overcom e this problem w e m odified the algorithm to include explicit orthogonalization of the conjugate-vectors at each iteration. W e called this alg o rith m the G ram -Schm idt least-sq u are-co n ju g ate-g rad ien t (GS-LSCG) m ethod. In all the cases w e have applied the GS-LSCG m ethod to, we have n e v e r ex p erien c ed a p ro b le m w ith its co n v erg en ce (D evayya an d W ingham , subm itted in 1992)

The decision w as m ad e to ru n w ith the GS-LSCG m eth o d an d to exam ine its rate of convergence for v a rio u s surface p a ra m eters and in cid en t w aves. We found th at the rate of convergence of the GS-LSCG m eth o d d ep en d s less u p o n a p a rticu la r v alu e of the RMS h eig h t and correlation-length, b u t m ore u p o n there ratio. This ratio is proportional to the RMS surface slope. We also found th at the size of the surface, w hich

determ ines the m atrix size N, does not affect the rate of convergence. This im portant, because the advantages of the conjugate-gradient m ethod then grow s w ith N.

The potential advantage of an iterative m ethod is th at the iteration can be stopped once a "good solution" has been found. To establish the point at w hich to truncate the iteration, w e exam ined the difference b etw een the scattered far-field pow er co m p u ted w ith the ite rate d so lu tio n for the surface current density and the scattered far-field pow er com puted w ith the so lu tio n o b ta in e d by LU d eco m p o sitio n . For th e su rfaces w e have considered, sm all errors in the surface current density are m ap p ed to small errors in the scattered far-field, even w h e n the scattered p o w er is small. The com putational issues w ere investigated in the light of this result, by com paring the CPU-tim es required b y the GS-LSCG m eth o d and b y LU decom position. We found th at w h en the RMS surface slope is sm all, or w h en N is very large, the GS-LSCG m eth o d determ ines a good solution w ith an ord er of m agnitude reduction in the com putation req u ired b y LU decom position.

From the o n set of the in v estig atio n o u r in te n tio n w as to exam ine w ave scattering for several incident fields. The m ajor disadvantage of the GS-LSCG m ethod is th at the m ethod is im plem ented for one incident field at a time. LU decom position on the other hand, allows the solution for any incident field to be directly obtained. We have p resen ted in this thesis a num erically robust conjugate-gradient m ethod for scattering problem s that re q u ire so lu tio n s for sev eral in c id e n t field s. The m e th o d u se s the in fo rm a tio n o b ta in e d in p rev io u s im p le m e n ta tio n s to d e te rm in e an initial-guess at the so lu tio n of the m atrix -eq u atio n for each ad d itio n al in cid en t field. Flow ever, for the cases w e have considered, the surface currents for different incident fields prove too d istinct for the m eth o d to p rovide any significant com putational advantage over LU decom position.

N evertheless, ou r w o rk on the num erical so lu tio n of the MFIE by the conjugate-gradient m ethod, is relevant to scattering problem s th at require solutions for a few incident fields, or w h en the size of the m atrix prohibits the use of direct solution m ethods.

To solve the MFIE num erically the integral m ust be truncated at some point. The scattering problem described by the truncated integral-equation is th at of a w ave scattered from a patch of surface. From a com putational stan d p o in t a small patch size is preferable. H ow ever, since it is hoped th at the norm alized incoherent scattered pow er com puted for an ensem ble of rough surface patches will apply to the infinite rough surface, the patch size m u st be large enough to accom m odate the average scattering properties of the infinite surface. We have placed the p o in t at w hich to truncate the MFIE in to a m athem atical context. The incoherent scattered pow er for an illum inated patch of surface w as presented as the integral of the w eighted autocorrelation-function of the random com ponent of the field scattered by each point of the surface. By representing the incoherent scattered pow er in this m anner, the factor determ ining the size of a patch w as identified as the separation required for this random process to decorrelate. We presented ex am p les of b a ck sc atterin g -au to c o rre latio n -fu n c tio n s for a perfectly- conducting, G aussian ro u g h surface. In the horizontal p olarization case, the autocorrelation-functions obtain a constant value w ith in a few surface correlation-lengths. This is also tru e of m ost of the cases for vertical polarization. The exceptions occur w h en the RMS surface height is small. For these geom etries the au to co rrelatio n -fu n ctio n s h av e an oscillatory c o m p o n e n t over the en tire le n g th of the fo o tp rin t. We su sp ect this p h enom enon is d u e to a surface w ave. The presence of a surface w ave com plicates the issue of w here to tru n cate the MFIE. H ow ever, for the surfaces w e have considered w e suspect th at the diffuse scattering of the surface w ave is relatively small. The results presented in this study provide evidence th at a relatively sm all patch size can accurately rep resen t the

second-order scattering properties of the infinite surface. This is possible because of the sm all correlation-length of the ran d o m com ponent of the scattering-function. In fact, w e consider th a t the lim it on the p atch size relates m ore to the m ethod used to reduce the scattering from the patch edges. The tap ered incident w ave used in o u r num erical sim ulations, for exam ple, is less consistent w ith the w av e eq u atio n as the tap e rin g is increased.

O ur attention th en centred o n the scattered far-field an d the expected value of the scattered pow er. The field scattered from a ro u g h surface is the scattered field obtained w ith the K irchhoff ap proxim ation p lu s the field due to the integral in the MFIE, w hich w e call the integral-field. In the high frequency lim it w ave scattering is not com plicated by diffraction, an d the role of the Kirchhoff and integral-fields are understo o d . In this lim it, the Kirchhoff-field is d u e to the single-reflection of incom ing rays from the surface. The integral-field is req u ired to account for sh ad o w in g of the surface, an d m ultiple-reflections at the surface boundary. We presented a p ro ced u re for obtain in g from the so lu tio n of the MFIE tw o physically distin ct corrections to the K irchhoff approxim ation. O n the assu m p tio n th at the coherence betw een the field due to single reflections and the field d u e to m ultiple-reflections is negligible, a correction for sh ad o w in g is d eterm in ed from the linear m ean-square estim ate of the integral-field in term s of the Kirchhoff-field. The error in the estim ate, w hich provides the seco n d c o rrectio n to th e K irchhoff m e th o d , satisfies th e coherence properties of the scattered field due to m ultiple-reflections.

A rm ed w ith these p ro ced u res w e set ab o u t ap p ly in g them to ou r num erical sim ulations of w ave scattering from G aussian ro u g h surfaces w here the RMS height and correlation-length are of the sam e order, and of the o rd er as the electrom agnetic w avelength. We fo u n d th at for a RMS slope of 25° th ere is sm all difference b etw een the integral-field an d the linear, m ean-square estim ate of the integral-field in term s of the Kirchhoff-

field. Physically, w e su sp ect th at because th is difference is sm all, the scattered field d u e to the illu m in a tio n of th e surface by scattered , w avefronts is small too. We suspect th at for these surfaces the correction to the K irchhoff approxim ation p ro v id ed by the integral in the MFIE is for p artial-shadow ing and the diffraction by the surface of the incident and scattered waves. The discussions for surfaces w ith m oderate slopes centred on the relationship betw een the polarization of the incident w ave, and the degree of shadow ing at the surface. We consider that both the near and far- field resu lts p ro v id e stro n g evidence th a t th e degree of sh ad o w in g is sm aller for vertical polarization th an for horizontal polarization. C ontour- plots of the electrom agnetic field in the vicinity of the surface b o u n d ary w ere used to illustrate this point in the near-field. In the far-field, w e have found th at the average scattered pow er in the horizontal polarization case is b e tte r d escribed by the K irchhoff m eth o d w h e n th e correction for sh ad o w in g d eriv ed in (W agner, 1967) is u sed . The resu lts for vertical polarization o n the other h an d , show how the K irchhoff m eth o d gives a b etter estim ate of the b ack w ard scattered p o w e r w h e n the sh ad o w in g correction is not used.

The resu lts for G au ssian ro u g h surfaces w ith v ery large slopes illustrate the enhanced backscattering rep o rted in the literature (O'Donnell and M endez, 1987). In contrast to the results for a RMS slope of 25°, for a RMS slope of 45° the difference betw een the integral-field an d the estim ate of the integral-field in term s of the Kirchhoff-field is not small. Physically, w e suspect this difference is due to the illum ination of the surface by waves scattered from other parts of the surface.