Like and cross polarized scatter cross sections for two dimensional, multiscale rough surfaces based on a unified full wave variational technique

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1 RADIO SCIENCE, VOL. 46,, do:10.109/010rs004441, 011 Lke and cross polarzed scatter cross sectons for two dmensonal, multscale rough surfaces based on a unfed full wave varatonal technque Ezekel Bahar 1 Receved 7 May 010; revsed December 010; accepted 15 Aprl 011; publshed July 011. [1] A varatonal method s used to select the specfc, smooth decomposton of the total surface heght spectral densty functon nto surface heght spectral densty functons for the larger and smaller scale surfaces. Usng ths decomposton, the total lke and crosspolarzed scatter cross sectons are expressed as weghted sums of physcal optcs scatter cross sectons assocated wth the larger scale surfaces and the tlt modulated scatter cross sectons for the smaller scale surfaces. Ths varatonal technque has been shown to be statonary over a wde range of the varatonal parameter. Snce only the slopes of the largerscale surfaces tlt modulate the cross sectons of the smaller scale surfaces, t s necessary to select surface heght spectral densty functons for the larger scale surfaces that do not requre the ntroducton of artfcal spatal cutoff wave numbers for the spectral densty functons. The methods used to smoothly decompose the surface heght spectral densty functons result n no artfcal rapd fluctuatons n the correspondng surface heght autocorrelaton functons for the smaller and larger scale surfaces. Ths method can be appled to the remote sensng of rough sea or land surfaces. Ctaton: Bahar, E. (011), Lke and cross polarzed scatter cross sectons for two dmensonal, multscale rough surfaces based on a unfed full wave varatonal technque, Rado Sc., 46,, do:10.109/010rs Introducton [] Usng a unfed full wave approach, the scatter cross sectons for rough surfaces are expressed as weghted sums of cross sectons for the larger and smaller scale surfaces [Bahar, 1981a]. The contrbutons of the larger scale surfaces (for whch the rad of the curvature are larger than the electromagnetc wavelength) are the physcal optcs [Beckmann and Spzzchno, 1963] cross sectons reduced by a factor equal to the characterstc functon squared of the smallerscale surface. The contrbutons of the smaller scale surfaces are tlt modulated by the slopes of the larger scale surfaces only [Valenzula, 1968; Bahar et al., 1983a, 1983b; Bahar and Kubk, 1993]. The specfc decomposton of the total surface heght spectral densty functon s consdered n detal n secton 4. Assocated wth ths decomposton s a varatonal parameter equal to the rato of the mean square heghts of the larger scale surface and the total rough surface (n = hh l /hh ). The statonarty of the solutons for the total scatter cross sectons s the bass for the varatonal method descrbed n ths paper. For the llustratons, both Gaussan [Brown, 1978] spectral densty functons and Person Moskowtz [Barrck, 1974] spectral densty functons are consdered n detal. The decomposton s conducted n a smooth contnuous manner n order to avod artfcal rapd fluctuatons n 1 Electrcal Engneerng Department Unversty of Nebraska Lncoln, Lncoln, Nebraska, USA. Copyrght 011 by the Amercan Geophyscal Unon /11/010RS the correspondng surface heght autocorrelaton functons [Bahar and Kubk, 1993]. These artfcal rapd fluctuatons occur when the decomposton s performed by selectng a specfc spatal wave number k d that separates the spectral densty of the larger and smaller scale surfaces. Ths results n dscontnutes of the ndvdual spectral densty functons. Snce the surface heght autocorrelaton functons of the ndvdual surfaces are the Fourer transforms of the correspondng surface heght spectral densty functons, t s these dscontnutes that result n the artfcal rapd fluctuatons n the correspondng autocorrelaton functons. [3] Of specal nterest, n performng the decomposton, s the necessty to have fnte means square slopes for the largerscale surfaces. For the Person Moskowtz spectral densty functon ths has usually been acheved by lmtng the spectrum of the larger scale surface on selectng an artfcal cutoff spectral wave number k c [Brown, 1978] as the upper lmt of the Person Moskowtz spectrum. Ths results n another dscontnuty for the assumed spectral densty functon. More recently, emprcal methods have been adopted to modfy the Person Moskowtz spectral densty functon for large values of the spatal wave number k, such that the correspondng mean square slope remans fnte. The method presented n ths paper to decompose the total spectral densty functon does not requre the selecton of a cutoff spatal wave number k c, or the adopton of any other emprcal method to overcome ths problem. Snce we consder two dmensonal multscale rough surfaces n ths paper, both lke and crosspolarzed cross sectons are consdered. [4] Prelmnary numercal smulatons of the varaton technque for evaluatng the scatter cross sectons of multple 1of1

2 scale rough surfaces have been conducted for perfectly conductng meda below the nterface [Bahar and Crttenden, 008]. Furthermore, surfaces that are rough only n one dmenson y = h(x) were consdered. The plane of ncdence was assumed to be perpendcular to the x axs; thus, the crosspolarzed cross sectons were zero. In ths paper the meda below the two dmensonally rough nterfaces are characterzed by ther complex permttvty and permeablty. The tangental components of the electrc and magnetc felds are contnuous at the rough nterface where exact boundary condtons are mposed. The scatter cross sectons are shown to be statonary over a broad range of the varatonal parameter n. It has also been shown that snce the horzontally polarzed cross sectons are much more senstve to tlt modulatons, the rato of the vertcally to the horzontally polarzed backscatter cross secton are near unty for near grazng angles of ncdence [Colln, 199, 008]. Ths has been as observed n feld measurements [Cloude and Coor, 00]. [5] The paper s organzed as follows: In secton, a survey of the related publcatons s presented. An overvew of the full wave method s gven n secton 3. In secton 4, a descrpton of the varatonal technque, ncludng formulaton of the problem, the analyss, and the prncpal results are presented. The mplementaton of the varatonal technque and the crtera for determnng the statonarty of the results are dscussed n secton 5. References to numercal smulatons are presented n secton 6. In secton 7, future work on numercal smulatons and practcal applcatons to remote sensng of rough sea surfaces (to determne wnd speed) and rough land surfaces (to determne sol mosture content for example) are brefly dscussed. Concludng remarks are gven n secton 8.. Comparsons Between Solutons for Rough Surface Scatter Cross Sectons Based on Geometrc Optcs, Physcal Optcs, the Standard Hybrd Two Scale Solutons, the Inverson of an Integral Equaton, and the Unfed Full Wave Technque [6] The study of rough surface scatterng has several applcatons n remote sensng [Barrck, 1970, 197]. Solutons for the scatter cross sectons of rough surfaces wth multple scales of roughness have been publshed [Hagfors, 1966; Burrows, 1973; Barrck and Peake, 1968; Valenzuela, 1968; Wrght, 1968]. Several of them are based on a hybrd physcal optcs perturbaton approach [Brown, 1978]. Usng ths approach, the smaller scale surfaces are depcted as f they were supermposed upon the larger scale surfaces (see Fgures 1a and 1b). To ths end the surface heght spectral densty functons for the entre surfaces are decomposed nto spectral densty functons for the smaller scale and the largerscale surfaces. The perturbaton approach [Rce, 1951] has been appled to obtan the solutons assocated wth the smaller scale surfaces whle the physcal optcs approach [Tyler, 1976; Beckmann and Spzzchno, 1963; Beckmann, 1968] has been appled to obtan the solutons assocated wth the larger scale surfaces. These two solutons are combned n an ad hoc manner to obtan the hybrd two scale solutons for the scatter cross sectons of the entre rough surface [Brown, 1978]. Detaled comparsons between these dfferent methods have been publshed by Barrck and Peake [1968] and Bahar [1987]. The prncpal problem wth the hybrd, ad hoc approach s that the soluton crtcally depends upon the manner n whch the spectral splttng s accomplshed [Bahar and Barrck, 1983]. Furthermore, ths approach apparently contradcts conservaton of energy, snce the larger scale rough surfaces cannot scatter more power by smply supermposng smaller scale rough surfaces upon them. A comparson between the full wave soluton presented here and a soluton based on the nverson of an ntegral equaton has been conducted [Colln, 199]. The unfed full wave approach descrbed n secton 4, on the other hand, s gven n terms of a weghted sum of two cross sectons [Bahar, 1981a; Bahar and Ftzwater, 1984, 1985; Bahar et al., 1983a, 1983b]. The contrbuton assocated wth the larger scale surface s reduced by a factor equal to the magntude squared of the characterstc functon for the smaller scale surface, whle the cross secton, assocated wth the smaller scale surface s tlt modulated by the slopes of the larger scale surface. Thus, dependng on the specfc spectral splttng assumed, as the contrbuton assocated wth the smaller scale surface s ncreased, the contrbuton assocated wth the larger scale surface s decreased and vsa versa. Ths feature of the unfed full wave approach makes the soluton statonary over a very wde range of the varatonal parameter assocated wth ths technque. The detals of ths varatonal technque are descrbed secton Overvew of the Full Wave Solutons and Formulaton of the Problem [7] The overvew of the full wave solutons starts wth the formal solutons to the generalzed telegraphsts equatons [Schelkunoff, 1955] for electromagnetc wave scatterng from two dmensonally rough surfaces. These generalzed telegraphsts equatons are sets of coupled dfferental equatons for the forward and backward propagatng wave ampltudes. Schelkunoff s method leadng to coupled dfferental equatons for the wave ampltudes s dstnct from ntegral equaton methods. The generalzed telegraphsts equatons are derved from Maxwell s equatons upon substtutng complete expansons for the electromagnetc felds. They consst of the radaton term, the lateral waves and the guded surface waves [Bahar, 1973a, 1973b]. In ths paper the scattered far felds are of partcular nterest; thus, only the radaton term s presented n detal. Snce the expansons of the felds are n the vertcal planes (x = const), exact boundary condtons are mposed at the rough nterface between free space (wth permttvty " 0 and permeablty m 0 and the medum below the nterface wth permttvty " and permeablty m) (see Fgures 1a and 1b). [8] For wavegudes wth fnte cross sectons and perfectly conductng rough boundares consdered by Schelkunoff, the complete expanson of the felds conssts of a dscrete set of propagatng and evanescent wavegude modes. For the problem consdered here the complete expansons conssts of a branch cut ntegral assocated wth the radaton felds and a branch cut ntegral assocated wth lateral waves and guded surface waves assocated wth the poles of the Fresnel reflecton coeffcents. Couplng between these three speces of the full wave expansons have been consdered n detal [Bahar, 1977]. They are of partcular nterest for recevers of1

3 Fgure 1. (a) Rough surface y = h(x, z), wave vectors and angles of ncdence and scatter n the fxed coordnate system. (b) Rough surface y = h(x, z), wave vectors and angles of ncdence and scatter n the local coordnate system. near the nterface. Lateral waves and surface waves are not excted when the surface s perfectly conductng. [9] In Fgure 1a the two dmensonally rough surface fðx s ; y s ; z s Þ ¼ y s hx ð s ; z s Þ ¼ 0 L x s L; l z s l; hx ð s ; z s Þ ¼ 0; jx s j L; jz s j l; ð1þ s represented n the fxed coordnate system wth unt vectors a x, a y and a z, and n, n f are unt vectors n the drectons of the ncdent wave and scattered waves above the nterface. In Fgure 1b, the rough surface s represented n the local coordnate system, wth n normal to the larger scale surface. For the smaller scale surface, the correlaton length (or the radus of curvature) s smaller than the wavelength l of the electromagnetc exctaton. The two other orthogonal unt vectors of the local coordnate system le n the local tangent plane (see Fgures 1a and 1b). [10] The frst order teratve solutons to the telegraphsts equatons are the felds mpressed on the surface, wth wave couplng neglected. The second order teratve solutons take nto account wave couplng [Bahar, 1981b, 1987; Bahar and Rajan, 1979]. Colln [199] refers to these solutons as Bahar s orgnal full wave solutons. Hgher order solutons, whch take nto account multple scatter, are shown to be assocated wth enhanced backscatter [Bahar and El Shenawee, 001]. Solutons, based on the nverson of ntegral equatons, are show to be n total agreement wth the orgnal full wave solutons [Colln, 199]. The orgnal full wave solutons are applcable to rough surfaces wth small mean square slopes. However, unlke Rce s [1951] smallperturbaton solutons, applcable to rough surfaces wth RMS heghts and slopes of the same order of smallness, the orgnal full wave solutons are not restrcted to rough surfaces wth small mean square heghts. On applyng a unfed full wave, two scale model of the rough surface, the scatter cross sectons are expressed as weghted sums of physcal optcs cross sectons for the larger scale surfaces and cross sectons for the smaller scale surfaces that are tlt modulated by the slopes of the larger scale surfaces only, snce the 3of1

4 slopes of the smaller scale surfaces do not contrbute to tlt modulaton. The slopes of the larger scale surfaces need not be small as wth the orgnal full wave soluton. 4. Descrpton of the Full Wave Varatonal Technque 4.1. Formulaton of the Problem [11] Tradtonal analytcal technques to determne the lke and cross polarzed scatter cross sectons for rough surfaces are classfed as physcal optcs solutons [Beckmann and Spzzchno, 1963] and perturbaton solutons [Rce, 1951]. The physcal optcs solutons are generally restrcted to rough surfaces wth large rad of curvature compared wth the electromagnetc wavelength. Physcal optcs scatterng occurs prmarly n the neghborhoods of the statonary phase, specular ponts on the surface where the normal to the surface bsects the wave vectors assocated wth the ncdent and scattered waves (see Fgures 1a and 1b). The perturbaton solutons are generally restrcted to smaller scale surfaces wth mean square heghts of the order of the electromagnetc wavelength or less. [1] For good conductng surfaces, the physcal optcs solutons are practcally ndependent of the polarzaton of the waves. However, the perturbaton solutons are strongly dependent on polarzaton, partcularly for backscatter at near grazng ncdence. Very often natural rough surfaces do not satsfy the restrctons assocated wth the physcal optcs solutons or the perturbaton solutons. It s therefore necessary to consder two scale models of the rough surfaces [Hagfors, 1966; Burrows, 1973; Valenzuela, 1968; Wrght, 1968; Barrck and Peake, 1968; Brown, 1978] for problems of scatterng from composte rough surfaces. These solutons requre the decomposton of the surface heght spectral densty functons nto ndvdual spectral densty functons for the larger and smaller scale rough surfaces. Solutons based upon an ad hoc, hybrd, physcal optcs perturbaton approach, are shown to be very senstve to the specfc decomposton of the surface heght spectral densty functons for the entre surface nto spectral densty functons for the larger and smaller scale surfaces [Brown, 1978].The unfed full wave solutons presented here have been shown to reduce to the physcal optcs solutons [Beckmann and Spzzchno, 1963] n the hgh frequency lmt and to the small perturbaton soluton [Rce, 1951] n the low frequency lmt. Moreover the full wave solutons based on a two scale model of the rough surface [Bahar, 1981a] are expressed as weghted sums of larger and smaller scale surfaces. Ths feature of the unfed full wave solutons contrbutes sgnfcantly to the statonarty of the varatonal technque presented here. 4.. Prncpal Advantages of the Full Wave Varatonal Technque [13] The major advantages of the novel unfed full wave varaton technques over the standard hybrd two scale solutons are summarzed below. [14] 1. The decompostons of the surface heght spectral densty functons are performed n a smooth contnuous manner (rather than a dscontnuous manner [Brown, 1978]) such that there are no non physcal rapd fluctuatons n the correspondng surface heght autocorrelaton functons [Bahar, 1969; Bahar and Kubk, 1993]. [15]. The slopes of the larger scale surfaces whch modulate the scatter cross sectons of the smaller scale surfaces are fnte, wthout ntroducng an artfcal cutoff spatal wave number to the Pearson Moskowtz spectral densty functon [Brown, 1978]. [16] 3. The total scatter cross sectons are expressed as weghted sums of the scatter cross sectons for the larger and smaller scale rough surfaces [Bahar, 1981a]. They are not summed up n an ad hoc manner [Brown, 1978]. Ths weghted sum s consstent wth energy conservaton. It also makes the total scatter cross sectons statonary over a wde range of the varatonal parameter (the rato of the mean square heghts of the larger scale surface and the total surface) [Bahar and Lee, 1994; Bahar and Crttenden, 008]. [17] 4. The solutons for the scatter cross sectons for the smaller scale surfaces are not lmted to surfaces wth small mean square heghts compared wth the electromagnetc wavelength, as s the case when small perturbaton solutons are used [Rce, 1951]. [18] 5. The scatter cross sectons are polarzaton dependent [Beckmann and Spzzchno, 1963; Bahar, 1981b; Bahar and Lee, 1994]. [19] 6. These full wave solutons explan why the cross sectons for the horzontally polarzed waves are much more senstve to tlt modulaton than the cross sectons for vertcally polarzed waves [Bahar and Kubk, 1993; Bahar et al., 1983a, 1983b; Bahar and Crttenden, 008]. [0] 7. These solutons explan why the ratos of the horzontally polarzed to vertcally polarzed cross sectons can approach unty even at near grazng angles, contrary to predcton based on the small perturbaton approach [Colln, 199; Cloude and Coor, 00]. [1] 8. Numercal smulatons based on these Full Wave analytcal solutons can be conducted much more rapdly than those based on averagng over many Monte Carlo smulatons of rough surfaces Full Wave Analyss [] For plane wave, lnearly polarzed exctatons the second order lnearly polarzed scattered felds are expressed n matrx form as [Bahar and Lee, 1994] G f S ¼ k ZZ 0 Sx p x h x þ S z p z h z p y v y exp k 0 r dk 0y dk 0z expðv r s Þdx s dz s G ; ðþ k 0x where G f S and G are l matrces, whose elements are the vertcally, V, and horzontally, H, polarzed components of Pf the scattered and ncdent felds E S and E P (P = V, H), respectvely. The ntegraton s over the surface varables x s and z s as well as the wave vector varables k 0y, k 0z for the radaton felds. It s assumed here that the mean plane of the rough surface s y = hh(x s, z s ) =0. [3] The surface element scatterng matrx S u s gven by [Bahar and Lee, 1994] S u k ; k ¼ cos 0 cos 0 R u k ; k ðu ¼ x; zþ: ð3þ 4of1

5 The elements of the matrx R u n (3) are R VV u ¼ rc 1 C1 C u S 0 S0 ð 1 1="r Þþð1 r ÞC u ðc 0 þ r C 1 Þ C0 þ rc1 ; ð4þ u ¼ " rc 1 C1 C u S 0 S0 ð 1 1=r Þþð1 " r ÞC u ðc 0 þ C 1 = r Þ C0 þ C 1 = ; ð5þ r R HH u ¼ D un r ð1 1= r ÞC 1 ð1 1=" r ÞC1 ðc 0 þ C 1 = r Þ C0 þ rc1 ; ð6þ R HV u ¼ D un r ð1 1=" r ÞC 1 ð1 1= r ÞC1 ðc 0 þ r C 1 Þ C0 þ C 1 = : ð7þ r R VH The wave vectors n the scatter and ncdent drectons are and k 0 k 0 n ¼ k 0 sn 0 cos a x þ cos 0 a y þ sn 0 sn a z k 0 k 0n ¼ k 0 sn 0 cos a x cos 0 a y þ sn 0 sn a z : ð9þ In (9), 0 s the elevaton angle (measured from y axs) and s the azmuth angle (measured from x axs toward the z axs). ð8þ C x ¼ S 0 cos þ S0 cos S 0 cos þ S0 ; ð10þ cos C z ¼ S 0 sn þ S0 sn S 0 sn þ S0 ; ð11þ sn p x v x D x p y v y ¼ sn cos S 0 S 0 sn cos ; p z v z D z p y v y ¼ sn cos S 0 S 0 sn cos : ð1þ In (4) (7), " r = " 1 /" 0, m r = m 1 /m 0, n r =(" r m r ) 1/ and h r = (m r /" r ) 1/ are the relatve permttvty, permeablty, refractve ndex and ntrnsc mpedance of the medum below the rough nterface (1). The snes and cosnes of the angles 0 and 1 above and below the rough nterface (denoted by S 0, S 1 and C 0, C 1 ) are related through Snell s law. [4] The vectors p and are gven n terms of the ncdent k 0 and scatter k 0 wave vectors. The subscrpts 0, 1 are for the medum 0(y > h (x s, z s )) and medum 1(y < h (x s, z s )), respectvely: p ¼ k 0 þ k 0 ; v ¼ k 0 k 0 ; ð13þ On ntegratng () by parts, t s expressed as [Bahar and Lee, 1994] G f S ¼ k ZZ 0 ½ ð S k ; k exp k0 r p x v x þ p z v z p y y exp v r sþexpðv r t Þ dk 0y dk 0z dx s dz s G ¼ G f G f D v y k : 0x ð15þ In (15) t s assumed that h(x s, z s ) vanshes for x s > L and z s > l, (1), and r t ¼ x s a x þ z s a s ¼ r s ha y : Snce p v ¼ 0, t follows that ð16þ ðp x v x þ p z v z Þ=p y v y ¼ 1: ð17þ The expresson for the surface element scatterng matrx S( k, k ) s gven by (4) (7) wth excepton that C u and D u are replaced by cos( ) and sn( ), respectvely. [5] The solutons presented n (15) are referred to by Colln [199] as the orgnal full wave solutons. Colln [199] uses a dfferent full wave approach to the problem of scatterng of plane waves from perfectly conductng surfaces based on an nverson of an ntegral equaton. Colln s results are also shown to be n complete agreement wth the orgnal full wave results for the perfectly conducng case ( " r ), m r =1). f [6] The second term G D n (15) can be ntegrated wth respect to x s and z s. For L and l, the ntegratons yeld the Drac delta functons d(v x )d(v z ). Thus, ths f term G D reduces to the specularly reflected plane wave. The full wave soluton G f (15) ncludes the dffusely scattered feld G f S, as well as the specularly reflected feld G f D. [7] When the observaton pont s at a very large dstance from the rough surface (k 0 r k 0 L 1 and k 0 r k 0 l 1), the ntegraton wth respect to the scatter wave vector varables k 0y, k 0z can be performed analytcally usng the statonary phase method. Thus, f the observaton pont s n the drecton n f r= jj¼ r sn f 0 cos f a x þ cos f 0 a y þ sn f 0 sn f a z ; ð18þ the dffuse far felds scattered from the rough surface are [Bahar and Lee, 1994] Z l Z L G f S ¼ G 0 S k f ; k ½ exp ð v rs Þexpðv r t ÞŠdx s dz s G l L ¼ G f G f D : ð19þ The expressons for S( k f, k ) n (19) are the same as the expressons S( k, k ) n (15) except that the scatter wave vector k s replaced n k f. Thus, k f 0 = k 0 n f and k f 1 for y < h (x s, z s ), are related to k f 0 through Snell s law. Furthermore, and r s and r are poston vectors from the orgn to ponts on the rough surfaces and to the observaton pont, respectvely: r s ¼ x s a x þ ha y þ z s a z ; r ¼ xa x þ ya y þ za z ¼ rn f : ð14þ and v ¼ k 0 n f n ¼ vx a x þ v y a y þ v z a z ð0þ G 0 ¼ k0 exp ð k 0rÞ=v y r: ð1þ 5of1

6 In (0), v y = k 0 (C 0 + C f 0 )=k 0 (cos 0 + cos f 0 ) When the ntegratons wth respect to x s and z s are performed, the term G f D s shown to be the flat surface quas specular (zero order) scattered feld whch s proportonal to (4Ll/v x Lv z l)sn v x L sn f v z l. The expresson for the quas specular scatter term G D s the same as the expresson for the total feld G f except that r s n G f f s replaced by r t n G D (equaton (16)). Thus, for h(x s, z s ) = 0, they are dentcal and G f s =0. [8] For surfaces wth small Raylegh roughness parameters b =4k 0 hh 1, the exponent exp(v y h) appearng n G f s expanded n a Taylor seres. In ths case the frst term n the ntegrand of (19), whch s proportonal to v y h, s precsely equal to the frst order small perturbaton soluton [Rce, 1951]. [9] To remove the small slope assumpton used to derve the teratve orgnal full wave soluton (19) (from the generalzed telegraphsts equaton), the surface element (dfferental) scatterng matrx S( k f, k ) (whch accounts for the sources nduced on the rough surface by the ncdent feld) s replaced by the followng scatterng matrx assocated wth the local coordnate system [Bahar, 1987]: S k f ; k ) T f S n k f ; k T D k f ; k : ðþ The quantty S n s obtaned from S, n (3.) by replacng the unt vector normal to the mean surface, a y by the unt vector normal to the actual surface [Bahar and Lee, 1994] n ¼ h x a x þ a y h z a z = 1 þ h x þ h 1=: z ð3þ Thus, the angles of ncdence 0 and scatter f 0 wth respect to the fxed, reference coordnate system (a x, a y, a z ) are replaced by the angles of ncdence n 0 and scatter fn 0 n the local coordnate system (n 1, n = n, n 3 ) (see Fgures 1a and 1b). Furthermore, the fxed planes of ncdence and scatter (normal to n a y and n f a y ) are replaced by the local planes of ncdence and scatter (normal to n n and n f n). The surface element scatterng matrx S n s nvarant to coordnate transformatons. The scatterng coeffcents S n where (as n (4) the frst superscrpt P denotes the polarzaton of the scattered wave and the second superscrpt Q denotes the polarzaton of the ncdent wave) vansh for n n 0 and n f n 0, correspondng to self shadow. The transformaton matrx T transforms the vertcally and horzontally polarzed waves n the fxed (reference) coordnate systems to the correspondng vertcally and horzontally polarzed waves n the local coordnate system, whle the transformaton matrx T f transforms the vertcally and horzontally polarzed waves of the local coordnate system back to the reference coordnate system. [30] Note that for the specular drecton wth respect to the reference coordnate system. n f! n fs ¼ n n a y ay ; ð4þ and the matrx R n (3) reduces to R ¼ RV 0 0 R H ; ð5þ where R V and R H are the Fresnel reflecton coeffcents for the vertcally and horzontally polarzed waves at the specular, statonary phase ponts, where n! n s ¼ v= jj: v ð6þ At these specular ponts, the local angles of ncdence 0 s and scatter 0 fs are gven by [Bahar et al., 1983a, 1983b] and cos s 0 ¼ cos fs 0 ¼n n s ¼ n f n s ; ð7þ cos fs s 0 cos 0 ¼ 1 þ cos f 0 cos 0 f sn 0 sn 0 cos f : ð8þ Thus, on applyng the statonary phase method to evaluate the ntegral n the hgh frequency lmt n closed form, the full wave solutons reduce to the physcal optcs statonary phase soluton n the hgh frequency lmt [Beckmann and Spzzchno, 1963]. The correspondng expresson for R n (P Q) vansh at the specular ponts. [31] Unlke the soluton that employs the Krchhoff approxmatons for the surface felds nduced by the ncdent wave, the generalzed telegraphsts equatons that satsfy exact boundary condtons are ntrnscally n agreement wth recprocty. [3] On retanng terms n frst order of smallness, the full wave soluton reduces to G f S ¼ G 0 Z l Z L l L S k f ; k vy h exp ð v rt Þdx s dz s G ; ð9þ Equaton (9) s precsely equal to the frst and second order small perturbaton soluton [Rce, 1951]. Thus, the same full wave expresson for the scattered felds presented here accounts for, n a unform, self consstent manner, (hgh frequency) specular pont scatterng [Beckmann and Spzzchno, 1963] as well as (low frequency) polarzaton dependent Bragg scatterng predcted by usng a small perturbaton approach. [33] The normalzed scatterng cross sectons for twodmensonal surfaces are defned as follows: D E 4r E Pf ¼ A y je Q j : ð30þ where A y s the projecton of the surface area (radar footprnt) onto the mean (reference) plane y =0. [34] The coherent scatterng cross sectons are defned as D E ¼ 4r E Pf C A y je Q j ; ð31þ and the ncoherent scatter cross sectons are defned as D E D E : ð3þ I For homogeneous rough surfaces, the surface heght autocorrelaton s only a functon of x d ( x s1 x s )andz d ( z s1 z s ). Furthermore, f the rad of curvature of the larger scale surfaces are assumed to be large compared wth the free space C 6of1

7 wavelength the slopes at pont may be approxmated by the value of the slopes at pont 1 h x h x1, h z h z1. [35] Full wave solutons for rough surface scatterng, wth heght slope correlatons ncluded, has been consdered [Bahar, 1991]. However, when the surface slopes are small, the surface heghts and slopes can be assumed to be uncorrelated and the full wave expresson reduces to [Bahar and Lee, 1994] D E I ¼ I Q ; ð33þ UC where the angle brackets denote statstcal average over the slopes h x, h z and I ðn Þ ¼ D I U n n U n f n P n f ; n jn =; ð34þ D where U( )P ( ) s Sancer s [1969] shadow functon. Furthermore, ZZ Q ¼ k4 0 v y h v y ; v y vy expðv x x d þ v z z d Þdx d dz d = n a y Z h v y ; v y vy J 0 ðv xz r d ¼ k4 0 v y ; Þr d dr d = n a y ð35þ where v xz = v v y. When the surface slopes (hh x 1, hh z 1) and the Ralegh roughness parameter b = 4k 0 hh 1 are of the same order of smallness, v y ; v y vy! exp v y h v y h C! v y h 1 h h : ð36þ where C s the normalzed surface heght autocorrelaton functon. Thus, Q k0 4 1 ZZ hh 1 h expðv x x d þ v z z d Þdx d dz d ¼ k0 4 Wðv x ; v z Þ; ð37þ where W(v x, v z ) s the two dmensonal surface heght spectral densty functon. For Gaussan autocorrelaton functons, Wðv x ; v z Þ ¼ h l c exp v x v z l c : ð38þ 4 When the surface slopes are assumed to be small, I can be expressed as D (h x, h z ) /p. Thus, on retanng frst order terms n (34), I ¼ 1 D ; ð39þ where D 0 s the zero slope value of D (equaton ()). Thus, 0 D E I ¼ I Q ¼ k0 4 SP D 0 Wðv x ; v z Þ: ð40þ [36] In the small heght/slope lmt the full wave soluton reduces to the small perturbaton soluton of Rce [1951]. Usng a statonary phase approxmaton of the full wave soluton, the most sgnfcant contrbutons to the scattered feld come from the neghborhood of the specular ponts on the rough surface where n! n s ¼ v jj v ¼ h xsa x þ a y h zs a z p ffffffffffffffffffffffffffffffffffffffffffffffffff h xs þ h zs þ 1 ; h xs ¼ v x ; h zs ¼ v z v y v y and ð41þ I! I ðn s Þ: ð4þ The product of I n (4) and Q n (35) yelds the physcal optcs soluton of Beckmann and Spzzchno [1963]: D E Z h I ¼ I ðn s Þ k4 0 PO v v y ; v y vy y J 0 ðv xz r d Þr d dr d : ð43þ In the hgh frequency lmt, t can be shown that ZZ v y ; v y jh x ; h z exp ½ vx ð x d þ v z z d ÞŠdx d dz d! 4 v y h x þ v x vy h z þ v z : ð44þ where d( ) s the Drac delta functon of the slopes at the specular ponts. Thus, n ths lmt, (40) reduces to the closed form geometrcal optcs soluton [Beckmann and Spzzchno, 1963] D E I ¼ 4k4 0 GO v 4 y D ðn s Þ P n f ; n =n s phxs ð ; h zs Þ; ð45þ where p(h xs, h zs ) s the slope probablty densty functon evaluated at the specular ponts, where n = n s (equaton (41)) Unfed Two Scale Solutons for Composte Random Rough Surfaces [37] For composte rough surfaces wth very large to very small scales of roughness (assocated wth the correlaton lengths or rad of curvature), nether the physcal optcs solutons nor the small perturbaton solutons are sutable for the evaluaton of the lke and cross polarzed scatter cross secton. The polarzaton dependent unform full wave solutons whch are not restrcted by the Raylegh roughness parameter b = k 0 hh have been shown to merge unformly wth the hgh frequency physcal optcs solutons and the low frequency small perturbaton solutons. Introducton of the local coordnate systems was used to remove the small slope restrcton of the orgnal full wave soluton. However, only the slopes of the larger scale surface need to be accounted for n determnng the normal n to the rough surface snce, the smaller scale surface does not satsfy the large rad of curvature crteron. [38] In ths secton a two scale unfed full wave soluton s presented to account for the contrbutons to the scatter cross secton from the larger and smaller scale surfaces [Bahar, 1981a]. 7of1

8 [39] Consder a composte random rough surface wth surface heght hx; ð z Þ ¼ h l ðx; zþþh s ðx; zþ; ð46þ where the subscrpts l and s, denote larger and smaller. There s no restrcton to the mean square heghts hh l and hh s ; however, t s assumed that l cl of the larger scale surface s larger than the correlaton length l cs of the smaller scale surface h s : l cl l cs : ð47þ Denote c l and c s as the characterstc functons of the largerand smaller scale surfaces and denote c and c s as the jont l characterstc functons for the larger and smaller scale surfaces. Snce the characterstc functons are related to the Fourer transforms of the surface heght probablty densty functons, jj ¼ l s l j s j ; ð48þ where c and c are the jont characterstc functon and characterstc functon of the total surface (equaton (46)). [40] Equaton (48) can be expressed as l s l j s h j ¼ l For dstances r d l cs l cl, l j s h j þ l s s j j : ð49þ s! j s j and l 1: ð50þ Thus, (49) can be approxmated by (see Appendx A) h jj ¼ l l j s h j þ s j j : ð51þ s The physcal nterpretaton of (51) s the larger scale surface contrbuton to the scatterng cross secton s dmnshed (due to superposton of the smaller scale surface upon the largerscale surface) by the factor c s < 1. Ths feature of the unfed full wave soluton s consstent wth conservaton of energy; total power scattered by the rough surface cannot ncrease by supermposng a smaller scale surface upon the larger scale surface. Snce the larger scale surface meets the large rad of curvature crteron, the contrbuton of the larger scale surface s gven by the physcal optcs soluton reduced by the characterstc functon squared of the smallerscale surface, whle the contrbuton of the smaller scale surface to the cross secton s tlt modulated by the slopes of the larger scale surface only. [41] In secton 5, the decomposton of the spectral densty functon nto spectral densty functons for the largerand smaller scale surface s consdered n detal. Ths decomposton results n the correspondng expresson for the total cross secton as a weghted sum of the physcal optcs contrbuton of the larger scale surface hs l and the tlt modulated contrbuton of the smaller scale surface hs s [Bahar, 1981a], D P ¼ s j j l E þ s : ð5þ The above expresson for the scatter cross sectons, n terms of weghted sums of cross sectons for the larger and smaller scale surfaces, was ntroduced n 1981 [Bahar, 1981a]. Ths feature of the unfed full wave solutons sgnfcantly contrbutes to the statonarty of the varatonal approach [Bahar and Crttenden, 008]. 5. Implementaton of the Procedure to Decompose the Surface Heght Spectral Densty Functons and the Crtera Used to Determne the Statonarty of the Solutons [4] Two dfferent surface heght spectral densty functons (Gaussan and Person Moskowtz) are consdered here snce the spectral decomposton nto spectral densty functons for the larger and smaller scale surfaces are acheved nto two dfferent ways. In both cases a smooth and contnuous decomposton of the spectral densty functons s performed. Ths s done n order to avod non physcal fluctuatons n the correspondng surface heght autocorrelaton functons, whch are the Fourer transforms of the spectral densty functons. [43] Intally, a two dmensonal Gaussan sotropc surface heght autocorrelaton functon R(r d ) s assumed for the entre surface: h hhrþ ð r d Þhr ðþ ¼ h exp ð rd =l c Þ ¼ h Rrd ð Þ; ð53þ where r d = x d + z d, hh s the mean square heght, and l c s the correlaton length. The correspondng surface heght spectral densty functon s WðkÞ ¼ 1 4 Z hhrþ ð r d Þhr ðþ dx d dz d ¼ h l c 4 exp k lc ; 4 ð54þ where k = k x + k z and the nverse Fourer transform of W(k)s the two dmensonal autocorrelaton functon, hhrþ ð r d Þhr ð d Þ ¼ Z WðkÞdk x dk y : ð55þ For the Gaussan spectral densty functon (54), the largerscale spectral densty functon W l, s defned frst. It has the same form as (54) except that hh s replaced by hh l, the mean square heght and l c s replaced by l cl the correlaton length. Furthermore, W l (0) = W(0) such that h l c ¼ h l l cl ð56þ Therefore, the spectral densty for the smaller scale surface s W s ðkþ ¼ WðkÞW l ðkþ ¼ 1 4 h l c exp k lc 4 h l l cl exp k lcl 4 : ð57þ As a result of the choce (55), W s (0) = 0. Note that all these spectral densty functons are postve real for all real values 8of1

9 of the varable k. Furthermore, the surface heght autocorrelaton functon for the smaller scale surface s h s Rs ðr d Þ ¼ h Rrd ð Þ h l Rrd ð Þ ¼ h h exp ð rd =l c Þ h h l exp ð rd =l cl Þ : ð58þ The correspondng mean square slopes for the total and the larger scale surfaces are ¼ Z WðkÞk dk x dk z ¼ 4 h =l c ð59aþ l ¼ 4 h l =l cl ¼ h l hh lc lcl 3 ¼ h l hh : ð59bþ The slopes of the smaller scale surface do not tlt modulate the scatter cross secton of the smaller scale surface. The probablty densty functon for the larger scale slopes that tlt modulates the cross sectons for the smaller scale surface s assumed to be Gaussan ph ð x ; h z Þ ¼ 1 exp h x þ h z =4 l ; ð60þ l where h x and h z are dervates of the surface heght of the larger scale surface wth respect to x and z. Lkewse, the shadow functon depends on the normal of the large scale surface n l as P k f ; k =n l ¼ U n n l U n f n l P n ; n f =n l : ð61þ Thus, the local angles of ncdence and scatter cannot be larger than p/. [44] For the Person Moskowtz surface heght spectral densty functon, t s the surface heght spectral densty functon for the smaller scale surface that s ntally chosen. It dffers from the surface heght spectral densty functon of the entre surface only n ts mean square heght and the value of k for whch the wnd generated surface heght spectral densty functon s maxmum, W MX. Thus, an sotropc Person Moskowtz surface heght spectral densty functon s gven explctly n terms of the mean square heght by WðkÞ ¼ h 6 k 4 ðk þ Þ 4 : ð6þ The correspondng surface heght normalzed autocorrelaton functon s, RðÞ¼ 1 þ 1 8 K 1 ðþ K 0 ðþ; ð63þ In (6), hh s the mean square heght. In (63) the K are modfed Bessel functons of orders 0 and 1 and the argument z = r d, where ¼ 335:V 4 1= ðcmþ 1 ; ð64þ and V s the wnd speed n m/s. The maxmum value of W(k) s for k = : ðk ¼ Þ ¼ h 3=8 : ð65þ W MX The surface heght spectral densty for the smaller scale surface s W s ðkþ ¼ h s 6 s k 4 k þ 4 ; ð66aþ s W s ðk ¼ s Þ ¼ h s 3=8 s ; ð66bþ where hh s s the mean square heght of the smaller scale surface and W s peaks for k =. We select hh s hh and hh s s = hh s such that W(k ) =W s (k s ). [45] Thus, W s (k) W(k) for all values of k. The correspondng expresson for the normalzed surface heght autocorrelaton functon for the smaller scale surface s the same as (11) except that z = s r d. [46] The surface heght spectral densty functon for the larger scale surface s W l ðkþ ¼ WðkÞW s ðkþ 0 ð67þ for all values of k. The correspondng mean square heght for the larger scale surface s h l ¼ h h s : ð68þ The correspondng normalzed surface heght autocorrelaton functon s R l ðþ¼r ðþr s ðþ: ð69þ Unlke the mean square slopes for Gaussan spectral densty functons (8), for the Person Moskowtz spectral densty functons the ntegral n (4.7a) wth the lmts (, ) s not fnte. Therefore t s common practce [Brown, 1978] to truncate the lmts of ntegraton to ( k c, k c ) where k c, the cutoff spatal wave number s usually chosen to be less than p/l and l s the wavelength of the electromagnetc exctaton. However, snce W l (k) defned n (68) vares as k 6 for k s the ntegraton n (59a) can be performed wth the lmts (, ) wthout choosng an artfcal cutoff spatal wave number, k c. [47] The mean square slope of the larger scale surface s l ¼ 6 h ln ð s =Þ ¼ 6ks h s ln ð s =Þ: ð70þ Ths s a very desrable feature of W l (l) snce the mean square slope wth the lmts ( k c, k c ) s senstve to the choce of k c. For the purpose of the analyss n ths paper, the mean square slopes of the total surface or the smaller scale surface are not needed snce only the slopes of the larger scale surface tlt modulate the scatter cross sectons of the smaller scale surface, not the mean square slope of the smaller scale surface. [48] The varatonal parameters n that s sutable to examne the statonarty of values of the scatter cross sectons s the rato of the mean square heght of the larger scale surface to the total mean square heght of the entre rough surface: n ¼ h l = h : ð71þ 9of1

10 Thus, for n = 1, the entre rough surface s regarded as largerscale and the total scatter cross sectons are gven by the hghfrequency physcal optcs soluton whch s obtaned from the full wave soluton through statonary phase ntegratons. Ths ntegraton results n the evaluaton of the surface element scatter coeffcents at the specular ponts on the surface, where the normal to the surface s along the vector v = k f k whch bsects the scatter and ncdent wave vectors. For n =0, the entre rough surface s regarded as smaller scale and the scatter cross sectons are gven by the orgnal full wave soluton wth no slope modulaton [Bahar and Rajan, 1979]. [49] For the values of 0 < n < 1, the scatter cross sectons are gven by the weghted sum of the larger scale (physcal optcs) cross secton (multpled by the factor equal to the absolute value of the characterstc functon squared c s of the smaller scale surface) plus the scatter cross secton for the smaller scale surface, tlt modulated by the slopes of the larger scale surface. Thus, as n decreases from n =1ton =0, the contrbuton of the smaller scale cross sectons ncreases and the contrbuton of the larger scale surface decreases due to the weghtng factor that multples the physcal optcs cross secton (snce c s decreases as n decreases). Ths bult n feature of the unfed full wave soluton results n the statonarty of the cross secton over a broad range of the varatonal parameter n. [50] The crteron used to determne the statonarty of the solutons for the backscatter cross secton s the followng norm of the error over, the full range of ncdent angles, from normal to near grazng [Bahar and Crttenden, 008]: E n Dn ¼ X PP nþdn 0 PP nþdn 0 PP þ PP n 0 n 0 h l h l nþdn þ h l nþdn h l! n ; n ð7þ where hs PP n+dn and hs PP n are values for the backscatter cross sectons correspondng to two consecutve values of mean square heghts of the larger scale surface (h ) n+ Dn and (h l ) n, respectvely. The norm of the error defned n equaton (7) s a dscretzed form of the followng ntegral: En ð Þ ¼ ln PP n; 0 d 0 : ð73þ In (7) the evaluated normalzed scatter cross sectons s PP (n, 0 ) depend on the varatonal parameter n and the angle of ncdence and K s constant. The statonarty of the results over a broad range of the varatonal parameter n, assocated wth the decomposton of the surface heght spectral densty functon W(k), manfests tself as a broad mnmum of norm of the error E(n) [Bahar and Crttenden, 008]. 6. Numercal Smulatons to Substantate the Varatonal Technque [51] The varatonal technque descrbed n ths paper has been substantated by conductng a seres of numercal smulatons for surfaces that are rough n one dmenson [Bahar and Crttenden, 008]. In these smulatons the scatter cross sectons for vertcally and horzontally polarzed waves were evaluated for values of the varatonal parameter n (equaton (71)) rangng from 0 to 1 at angles of ncdence from normal to near grazng. Rough surfaces wth a wde range of mean square heghts were consdered. The crtera used to determne the statonarty of the solutons descrbed n secton 5 (equaton (7)) were appled to these numercal smulatons. It was shown that the results were statonary over a very wde range of the varatonal parameter n rangng from 3 to 7. The norm of the error (7) was shown to be largest for values of n around zero and one, where n = 0 corresponds to regardng the entre surface as the smaller scale surface and n = 1 corresponds to regardng the entre surface as the largerscale surface. [5] These numercal smulatons also clearly demonstrate that for horzontally polarzed waves the cross sectons are strongly dependent on slope modulaton by the tlt of the larger scale surface for near grazng ncdence. However, for vertcally polarzed waves the dependence on slope modulaton s practcally nsgnfcant [Colln, 199; Bahar and Crttenden, 008]. These results are consstent wth expermental observatons that the rato of the two cross sectons s near one for grazng ncdence [Cloude and Coor, 00]. Ths s contrary to results based on the standard perturbaton soluton. 7. Future Work on the Proposed Varatonal Technque Appled to Two Dmensonally Rough Surfaces [53] Addtonal seres of numercal smulatons and expermental observatons are to be performed for surfaces that are rough n two dmensons pursuant to the analyss gven n ths paper. Surfaces that are characterzed by fnte conductvty, as descrbed n ths paper, also need to be consdered. [54] Applcatons to remote sensng of sea surfaces (to determne wnd speed for nstance) and land surfaces (to determne sol mosture content for nstance) are some of the practcal applcatons of ths analyss. 8. Concludng Remarks [55] The procedures leadng to varatonal solutons for the lke and cross lnearly polarzed scatter cross sectons for twodmensonally rough surfaces are presented n detal n ths paper. It s based on the use of unfed full wave solutons to express the total cross sectons as weghted sums of two cross sectons [Bahar, 1981a]. The frst, assocated wth the largerscale surface s the physcal optcs cross secton reduced by absolute value of the square of the characterstc functon for the smaller scale surface. The second s the scatter cross secton for the smaller scale surface that s tlt modulated by the slopes of the larger scale surface alone. Ths feature of the unfed full wave solutons [Bahar, 1981a] contrbutes sgnfcantly to the statonarty of the varatonal approach presented here. [56] The method for decomposng the scatter cross sectons for the entre surface heght spectral densty functon nto spectral densty functons for the larger and smaller scale surfaces s gven separately n detal n secton 5 for Gaussan and Person Moskowtz spectral densty functons, snce the procedures for the decomposton of these spectral densty functons are qute dstnct. [57] For entre surfaces wth Gaussan spectral densty functons, we frst choose Gaussan spectral densty functons 10 of 1

11 for the larger scale surface, such that the spectral densty functon for the smaller scale surface vanshes for the spatal wave number k = 0. Thus, when the total surface has a Gaussan spectral densty functon, hh l c = hh l l cl. Therefore for the varatonal parameter, n = hh l /hh 1, l cl l c. [58] For the total surfaces wth Person Moskowtz spectral densty functons we frst choose a Person Moskowtz spectral densty functon for the smaller scale surface such that W s (k ) =W(k ). In ths case therefore the varatonal parameter 1 n = hh s /hh = / s n whch the parameters s corresponds to (related to the surface wnd speed). In ths case the spectral densty functon for the larger scale surface has a fnte mean square slope, wthout ntroducng an artfcal cutoff spectral wave number. The correspondng surface heght autocorrelaton functons for the larger and smaller scale surfaces (the Fourer transforms of the surface heght spectral densty functons) do not possess artfcal rapd fluctuatons, snce the decompostons are acheved n a smooth, contnuous manner, rather than a dscontnuous manner. [59] For the prelmnary numercal smulatons, the scatter cross sectons are shown to be statonary over a wde range of the varatonal parameter n [Bahar and Crttenden, 008]. These results are also consstent wth observatons that the rato of the backscatter cross sectons for the horzontally and vertcally polarzed waves could be near unty for near grazng ncdence [Cloude and Coor, 00]. Ths s because the horzontally polarzed cross sectons are much more senstve to tlt modulaton than the vertcally polarzed cross sectons [Colln, 199; Bahar and Crttenden, 008]. Appendx A [60] The terms that were neglected by replacng (49) by (51) are expressed n ther Taylor seres expansons for r d l cs l cl as follows: 1 l In (A1), s j s j v y h l ½ 1 Rl ðr d l cl ÞŠj s j v y h s R s ðr d l cs Þ ða1þ R l ðr d l cl Þ 1 ; R s ðr d l cs Þ ð1=eþ and j s j < 1 ðaþ For 0 < r d < l cs l cl,(1 R l ) 0 and R s < 1; therefore, the contrbuton of (A1) to (49) s neglgble. [61] Acknowledgments. The author wshes to thank R. E. Colln for hs comments and suggestons regardng the varatonal approach. M. Crag, R. Odom, and L. Smth prepared ths manuscrpt. References Bahar, E. (1973a), Electromagnetc wave propagaton n nhomogeneous multlayered structures of arbtrary thckness Generalzed feld transforms, J. Math Phys. N. Y., 14(8), , do: / Bahar, E. (1973b), Electromagnetc wave propagaton n nhomogeneous multlayered structures of arbtrary thckness Full wave solutons, J. Math Phys. N. Y., 14(8), , do: / Bahar, E. (1977), Couplng between guded surface waves, lateral waves and the radaton felds by rough surfaces full wave solutons, Trans. Mcrow. Theory Tech., 5(11), , do: /tmtt Bahar, E. (1981a), Scatterng cross sectons for composte random surfacesfull wave analyss, Rado Sc., 16(6), , do:10.109/ RS016006p0137. Bahar, E. (1981b), Full wave solutons for the depolarzaton of the scattered radaton felds by rough surfaces of arbtrary slope, IEEE Trans. Antennas Propag., 9, , do: /tap Bahar, E. (1987), Revew of the full wave solutons for rough surface scatterng and depolarzaton: Comparsons wth geometrc and physcal optcs, perturbaton and two scale solutons, J. Geophys. Res., 9, , do:10.109/jc09c05p0509. Bahar, E. (1991), Full wave analyss for rough surface dffuse, ncoherent radar cross sectons wth heght slope correlatons ncluded, IEEE Trans. Antennas Propag., 39, , do: / Bahar, E., and D. E. Barrck (1983), Scatterng cross sectons for composte rough surfaces that cannot be treated as perturbed physcal optcs problems, Rado Sc., 18(), , do:10.109/rs01800p0019. Bahar, E., and P. Crttenden (008), Backscatter cross sectons of composte random rough surfaces based on the selecton of varatonal parameters to determne the spectral denstes of the larger and smaller scale surfaces, Int. J. Remote Sens., 9(), , do: / Bahar, E., and M. El Shenawee (001), Double scatter cross sectons for two dmensonal random rough surfaces that exhbt backscatter enhancement, J. Opt. Soc. Am. A Opt. Image Sc. Vs., 18(1), , do: /josaa Bahar, E., and M. A. Ftzwater (1984), Scatterng cross sectons for composte rough surfaces usng the unfed full wave approach, IEEE Trans. Antennas Propag., 3, , do: /tap Bahar, E., and M. A. Ftzwater (1985), Lke and cross polarzed scatterng cross sectons for random rough surfaces: Theory and experment, J. Opt. Soc. Am.,, , do: /josaa Bahar, E., and R. D. Kubk (1993), Tlt modulaton of hgh resoluton radar backscatter cross sectons, unfed full wave approach, IEEE Trans. Geosc. Remote Sens., 31, 19 14, do: / Bahar, E., and B. Lee (1994), Full wave solutons for rough surface bstatc radar cross sectons: comparson wth small perturbatons, physcal optcs, numercal, and expermental results, Rado Sc., 9, , do:10.109/93rs Bahar, E., and G. G. Rajan (1979), Depolarzaton and scatterng of electromagnetc waves by rregular boundares for arbtrary ncdent and scatter angles full wave solutons, IEEE Trans. Antennas Propag., 7, 14 5, do: /tap Bahar, E., C. L. Rufenach, D. E. Barrck, and M. A. Ftzwater (1983a), Scatterng cross secton modulaton for arbtrarly orented random rough surfaces, Rado Sc., 18(5), , do:10.109/rs018005p Bahar, E., D. E. Barrck, and M. A. Ftzwater (1983b), Computaton of scatterng cross sectons for composte surfaces and the specfcaton of the wave number where spectral splttng occurs, IEEE Trans. Antennas Propag., 31, , do: /tap Barrck, D. E. (1970), Rough surfaces, n Radar Cross Secton Handbook, pp , Plenum, New York. Barrck, D. E. (197), Remote sensng of sea state by radar, n Remote Sensng of the Troposphere, edtedbyv.e.derr,pp , U.S. Gov. Prnt. Off., Washngton, D. C. Barrck, D. E. (1974), Wnd dependence of quas specular mcrowave sea scatter, IEEE Trans. Antennas Propag.,, , do: / TAP Barrck, D. E., and A. Peake (1968), Revew of scatterng from surfaces wth dfferent roughness scales, Rado Sc., 3(8), Beckmann, P. (1968), The Depolarzaton of Electromagnetc Waves, Golem, Boulder, Colo. Beckmann, P., and A. Spzzchno (1963), The Scatterng of Electromagnetc Waves From Rough Surfaces, Pergamon, New York. Brown, G. S. (1978), Backscatter from a Gaussan dstrbuted perfectly conductng rough surface, IEEE Trans. Antennas Propag., 6, 47 48, do: /tap Burrows, M. L. (1973), On the composte model of rough surface scatterng, IEEE Trans. Antennas Propag., 1, 41 43, do: /tap Cloude, S. R., and D. Coor (00) A new parameter for sol mosture estmaton, paper presented at 00 IEEE Internatonal Geoscence and Remote Sensng Symposum and the 4th Canadan Symposum on Remote Sensng, Toronto, Ont., Canada. Colln, R. E. (199), Electromagnetc scatterng from perfectly conductng rough surfaces (a new full wave method), IEEE Trans. Antennas Propag., 40, , do: / Hagfors, T. (1966), Relatonshp of geometrc optcs and autocorrelaton approach to the analyss of lunar and planetary radar, J. Geophys. Res., 71, of 1