eq u atio n has a m uch w id er scope of application. There is considerable in te re s t in ite rativ e m eth o d s w ith in th e electro m ag n e tic sc atterin g co m m u n ity (Sarkar et al, 1981), (U m ashankar, 1988), (M arks, 1990). The procedures used to identify the divergence of the N eu m an n expansion, for exam ple, can be applied to N eum ann expansion of any Fredholm integral- equation (Baker, 1977) of the second-kind. Also, the w ork presented o n the conjugate-gradient m ethod, and in particular, avoiding ro u n d in g errors by u sin g G ram -S chm idt o rth o g o n alizatio n sh o u ld be of in te rest to those research ers w h o have fo u n d the initial convergence of th e conjugate- gradient m eth o d applied to there problem to be rap id , b u t th en diverges due to ro u n d in g errors.
In the num erical studies of ro u g h surface scattering the p o in t at w hich the integral-equations are truncated varies m arkedly from stu d y to study. G iven the problem of choosing a patch size large enough to preserve the second-order scattering properties of the infinite surface, b u t at the sam e tim e sm all e n o u g h to lim it th e co m p u tatio n al req u ire m e n t, w e have p laced the p o in t at w hich to tru n cate the in teg ral into a m athem atical context. O u r sim ulations su g g est th at a relativ ely sm all p atch size can describe the second-order scattering p ro p erties of the infinite surface. In fact, the lim it on the patch size appears to relate m ore to m eth o d u sed to g u ard edge effects. The m ethod w e have used to g u ard against these effects is to tap er the incident w ave to negligible levels at the p atch edges. This m ethod has also been used by Thorsos (1988), Thorsos and Jackson (1989), B roschat at at (1989) and m ost recently by Ish im aru an d C hen (1991). It w o u ld have been useful, if we h ad h ad the tim e, to com pare this approach w ith the m ethod for periodic gratings d u e to Jordan and Lang (1979). The integral-equation for a periodic grating is described along a closed contour o v er one p e rio d of the grating. The p erio d ic n a tu re of the surface is acco m m o d ated in to the G reens' fu n ctio n for th e sc atterin g problem . A p p lied to the ran d o m , ro u g h surface scatterin g problem , the ran d o m
ro u g h n ess is d efin ed over one p e rio d of the g ratin g . A lth o u g h the periodicity of the surface roughness m odulates the angular d istribution of scattered pow er by a p attern of interference fringes, the m eth o d of Jordan and Lang does not suffer from ''edge effects", because the integral-equation is bounded.
This thesis has com pared w ave scattering for b o th polarizations. The b u lk of the literatu re on w ave scattering is on the D irichlet scattering problem , w hich in the context of this stu d y is the horizontal polarization case. O ur results for horizontal polarization are consistent w ith the results p re se n te d by T horsos (1988). W e have also fo u n d th a t the K irchhoff approxim ation w h en u sed w ith the correction for sh ad o w in g d eriv ed in (W agner, 1967), gives a better description of the scattered p o w er as the surface correlation-length approaches the electrom agnetic w avelength. O ur w o rk has also d ealt w ith the vertical p o larizatio n case. H ere, w e have fo u n d th a t the back w ard scattered p o w er is b etter d escrib ed w ith the Kirchhoff m ethod w hen the correction for shadow ing is not used.
The results for a RMS slope of 25° show th at there is sm all difference b etw een the integral-field an d the lin ear m ean -sq u are estim ate of the integral-field in term s of the K irchhoff-field. A nalytic theories of w ave scatterin g , w ith the exception of the seco n d -o rd er K irch h o ff-iteratio n (Ishim aru an d C hen (1990 a, b), are a p p ro p riate w h en the RMS surface slope is less th an 25°. We suspect th at these theories operate in a region of the param eter space w here the scattered field due to the illum ination of the surface b y scattered, w avefronts is sm all. The n a tu re of the results for ro u g h surfaces w ith large slopes bare a strik in g resem blance to those presented by Ishim aru and C hen (1990 a, b), (1991) and Bruce and Dainty (1991). These authors have used the first tw o term s of a Kirchhoff iteration to describe w ave scattering from very rough surfaces in the region "SKI" of fig. 1*3. The first term of the iteration gives the expected scattered pow er
obtained w ith the shadow -corrected Kirchhoff m ethod. The second term is req u ired to account for enhanced backscattering. We have p resen te d a procedure for obtaining from the solution of the MFIE tw o corrections to the expected scattered pow er obtained w ith the K irchhoff approxim ation. This procedure has allow ed us to gain valuable insight into the scattering m echanism s th at operate at the surface boundary. We have fo u n d th at in m any cases the correction for sh ad o w in g is close to the correction for sh a d o w in g d eriv ed in (W agner, 1967). The second co rrectio n , w h ich physically w e suspect accounts for the illum ination of the surface by w aves scattered from other parts of the surface, for very rough surfaces is required to describe the an g u lar d istrib u tio n of the enhanced b ack w ard scattered power.