To solve the MFIE num erically the integral in (11) m u st be truncated at som e point. The scattering problem described b y the truncated integral- equation is th at of a w ave scattered from a p atch of surface. The p o in t at w hich the integral is truncated is im portant, because it is one of the factors th at d eterm in es the size of the m atrix u sed to rep resen t the MFIE, an d hence the com putation req u ired to determ ine the surface cu rren t density. From a c o m p u ta tio n a l s ta n d p o in t a sm all p a tc h size is p referab le. H ow ever, since it is h o ped th at the norm alized incoherent scattered pow er co m p u ted for an ensem ble of ro u g h surface p atch es w ill a p p ly to the infinite surface too, the patch size m u st be large enough to accom m odate the second-order scattering properties of the infinite surface.
The scattered far-field is obtained by integrating the field scattered by each p o in t o n th e surface b o u n d a ry . W e w ill refer to th e fu n ctio n d escribing the sp atial d istrib u tio n of scattered fields along the infinite surface as the scatterin g -fu n ctio n . F u rth erm o re, w e w ill use th e term "seatterin g -fu n ctio n " to refer to the specific case of the infinite surface illu m in a ted b y a u n ifo rm p lan e w ave. In this section, w e p re se n t an equation for the incoherent p o w er scattered from a w ide-sense-stationary, random ly, ro u g h surface as a function of the size of the illum inated area. W e w ill show th at the sep aratio n req u ired for the ran d o m com ponent of the scattering-function to decorrelate, is the factor determ ining the size of a patch.
plane wave is obtained from the integral
E»(m: e ‘,0®) = I J(m: x') K(m; 0®, x') dx'. (412)
H ere, ES(m: 0^, 0^) is the scattered field in the direction of 0^, for a uniform plane w ave incident at 0^, K(m: x') is the kernel of the scattered far-field in teg ral (41), or (4 2), an d J(m: x') is the surface c u rre n t den sity . The in teg ran d of (4-12) is the field scattered from the surface at x'. This w ill be rep resen ted by the function E^(m: x'), w hich w e refer to as the scattering- function. The d ep en d en ce of the scatterin g -fu n ctio n o n th e an g le of incidence and the scattering angle has been om itted from the notation. W e shall also take for g ran ted th at the scattered pow er has been norm alized w ith respect to the pow er incident on the surface.
The scattering-function
ES(m: x) = A(m: x) eikx(sin0‘ + sin0®>. (413)
is the p ro d u ct of a stochastic process and a determ inistic process (Ulaby et al, 1982). The determ inistic process, w hich is the com plex-exponential in (4-13), is due to the periodic phase m odulation of the incident w ave along the x-axis. The random com ponent A(m: x), m = 1 ,..., oo, describes the phase an d a m p litu d e m o d u la tio n of the scatterin g -fu n ctio n by the ra n d o m surface profile. The objective of this section is to determ ine the incoherent scattered pow er for an incident w ave
HÎ(x) = W(x)eik(xsin0'+ zcos0^) (414)
in term s of the ran d o m process A(m: x). In (4 14), W(x) is the fo o tp rin t of the incident w ave o n a flat surface. We shall assum e th at the affect of the
footprint W(x) is to linearly w eight the scattering-function along the x-axis. Based o n this assum ption, the far-field scattered from a surface illum inated by the tapered wave (4-14), is
ES(m) = W (x’) ES(m: x’) dx' (415)
I .
We w ill exam ine the validity of this assu m p tio n later. We w ill also m ake the assu m p tio n th at for w ide-sense stationary, ran d o m ro u g h surface the random process A(m: x), m = 1,..., «>, is w ide-sense stationary too (Ulaby et al, 1982). O n this assum ption, the expected scattered p o w er in term s of the scattering function is (see Papoulis, 1984)
i s as = R w (t) R(x) e'k (sin 8 + smG ) x dx (416) J - o o w h ere. R(x) = E [ A(x'+ X ), A*(x' ) 1 (417) a n d Rw(T)
= J
W(x'+ x) W '(x ') dx' . (418) J-ooH ere, R(x) is the autocorrelation-function of the stochastic process A(m: x'), m = 1, ..., oo, w hich we will call the scattering-autocorrelation-function.
A sim ilar expression for the coherent scattered pow er
f - i s
I Us I ^ = Rw(x) I HA 1 ^ eik(sine + smO ) x dx, (419) J - o o
o o
is d e riv e d u sin g (4 15). H ere, p.a isE [A (x)], w hich for a w ide-sense-
stationary process is by definition constant for all x (Papoulis, 1984). Finally, the incoherent scattered pow er (4 7) is obtained by subtracting (4 19) from (416),
i s
(fi = R w (i) ( R (T )-|p A |^ ) eik(sin8 + sinG ) z dx. (420)
J-oo
It can be recognized from (4-20) th at the effect of the footprint W(x) o n the incoherent scattered p o w er is to w eight ( R(x) - 1PA |^ ) by the fu nction R ^ (x ). Furtherm ore, for non-pathological, w ide-sense-stationary, random processes (Papoulis, 1984)
R(x) -> PA as X oo. (4 21)
If follow s from (4-21) th at if the scattering-autocorrelation-function (4* 17) obtains its asym ptotic value w ith in a separation very m uch sm aller th an the w id th of the illum ination, the finite w id th of the fo o tp rin t w ill have sm all affect on the incoherent scattered pow er. The separation required for the scattering-autocorrelation-function to obtain its asym ptotic value, is by definition the sep aratio n req u ire d for the the ran d o m com ponent of the scattering-function to decorrelate.
A resu lt th a t is im m ed iately available, is the in co h eren t scattered p o w er obtained w ith the Kirchhoff ap p ro x im atio n for a G aussian ro u g h su rface illu m in a te d b y a ta p e re d in c id e n t w av e. T he sc a tte rin g - a u to co rrelatio n -fu n ctio n o b tain ed w ith the K irchhoff a p p ro x im atio n is derived in (Ulaby et a l 1982),
c ( t ) = exp - ^ . (4 23)
L
H ere, a is the RMS surface height, Ç is the surface correlation-length and c(x) is th e n o rm a liz e d a u to c o rre la tio n -fu n c tio n of the surface. The asym ptotic value of is obtained by setting c(x) to zero, (w ith c(x) = 0, Rk(x) is the rough surface reflection coefficient discussed in § 51). It can be easily recognized from (4-22), th at Rk('^) obtains its asym ptotic value after a b o u t tw o surface co rrelatio n -len g th s. For an in cid en t w av e w ith a G aussian, footprint,
W(x) = (424)
an d
R w W = Rw (0) (4 25)
The fu nction Ry^ obtains its half pow er p o in t for x - 0*8y. Therefore, the the fo o tp rin t (4 24) w ill have sm all affect o n the n o rm alized incoherent sc attere d p o w er p ro v id e d the tap e rin g p a ra m eter y is several surface correlation-lengths long.
For the surfaces w e have considered, the K irchhoff app ro x im atio n is in ap p ro p riate an d the MFIE m u st be solved. In the next section w e w ill p re s e n t exam ples of the scatterin g -au to co rrelatio n -fu n ctio n s c o m p u ted from o u r num erical solutions of the MFIE.