Download - A numerical assessment of rough surface scattering theories: Horizontal polarization

Radio Science, Volume 27, Number 4, Pages 515-527, July-August 1992

A numerical assessment of rough surface scattering theories: Vertical polarization

Yunjin Kim, Ernesto Rodriguez, and Stephen L. Durden

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

(Received November 15, 1991; revised August 19, 1991; accepted August 20, 1991.)

In a previous paper, we presented a numerical evaluation of the regime of validity for various rough surface scattering theories. The evaluation was done for horizontal polarization by comparing these theories against numerical results obtained by the method of moments. In this paper we address the same issues for vertical polarization. The theories considered here are the small perturbation method, the Kirchhoff approximation, the momentum transfer expansion, the two-scale expansion, and the unified perturbation method (UPM). All these theories can be derived in a unified manner starting with the extinction theorem. The rough surfaces of interest are oceanlike surfaces, which exhibit height power law spectra. Considering both horizontal and vertical polarizations, the UPM provides best results among all theories considered.

1. INTRODUCTION

Even though there exists a host of approximate solutions, the regime of applicability of each rough surface scattering theory has not been well under- stood. The reason is that exact closed form solutions are not available for this problem. Recently, it has become more attractive to numerically solve the rough surface scattering problem. However, the numerical solution possesses several defects. First, it is technically difficult to solve three-dimensional scattering problems for large surfaces. Second, the numerical solutions lack the physical intuition which can be acquired from analytic expressions. The third defect is that numerical simulation does not allow a convenient way of performing the inversion of surface roughness, which is important for remote sensing. Hence it is important to have an approximate method appropriate for various natural surfaces. We are especially interested in applica- tions involving the ocean surface. Here we assume that if a method is accurate for describing the scattering of both horizontal and vertical polarizations, the method is also suitable for three-dimensional scattering. The horizontal polarization case has been presented by Rodriguez et al. [this issue]; the vertical polarization case is presented here. The approximations that we consider are the classical Kirchhoff [Beckmann and Spizzichino, 1963] and

Copyright 1992 by the American Geophysical Union.

Paper number 91RS02637. 0048-6604/92/91 RS -02637 $08.00

small perturbation method (SPM) (or Bragg scattering) approximations [Rice, 1951; Nieto-Vesperinas and Garcia, 1981], phase expansions [Shen and Ma- radudin, 1980; Winebrenner and Ishimaru, 1985a, hi, two-scale theories [Brown, 1978; McDaniel and Gor- man, 1983; Rodriguez, 1989], and momentum transfer expansion [Rodrœguez, 1989,1991]. These approximations are chosen since they can be derived from the extinction theorem in a unified manner.

The numerical comparison of rough surface scattering approximations with numerical solutions was initiated by Lentz [1974] and Fung and his collabo- rators [Axline and Fung, 1978; Fung and Chen, 1985; Chen and Fung, !988]. Recent contributions to the subject have been made by Thorsos and Jackson [1989], Thorsos [1988], Soto-Crespo et aI. [1990], and Sanchez-Gil and Nieto-Vesperinas [1991]. However, their comparisons are limited to the classical SPM and Kirchhoff theories and for surfaces whose correlation function is Gaussian. Here we examine in a unified way several recently developed approximations, in addition to the classical ones. The surfaces that we consider have a power law spectrum which is more representative of natural (or fractallike) surfaces such as the ocean or geological surfaces. The scattered field is calculated using the results presented by Jordan and Lang [1979] without invoking heuristic window functions.

In the next section we present the family of scattering theories under consideration. The third section provides a brief description of our humeri-

515

516 KIM ET AL.: NUMERICAL SCATTERING FROM ROUGH SURFACES

cal simulations and results. The final section sum- solve (5) perturbatively by expanding the source marizes our conclusions. function in a series whose order depends on the

power of a smallness parameter. Two broad classes of series expansions are considered: field and phase 2. BRIEF REVIEW OF SCATTERING THEORIES expansions. In the field expansion the source func.

In this section we establish our conventions and tion is assumed to be of the form review the ideas behind the various perturbation

approaches from a unified perspective. Consider a • • n vertically polarized electromagnetic wave incident f(x) = D(x) 24n)(x) n-? (6) on a perfectly conducting rough surface whose n•-0 mean is the x-y plane and whose height above the where D(x) is a function which is independent of the plane is given by h(x); that is, the surface does not smallness parameter. In the phase expansion, on change along the y direction, Assuming harmonic the other hand, the source function is assumed to be time dependence e-iø•t, the incident field is assumed of the form to be a plane wave of wavelength A = 2•r/k and

wave vectør k(i)^"(i)•kz(i): [•=• 1 œn

H(i)(x, z) = 9H(i)(x, z) = :9 exp [ikx(i)x - ik?z] (1) f(x) = D(x) exp #(n)(x) •.• . (7) I

In general, the function kz(kx)is defined by It is clear that by expanding this equation and kz(kx) = (k 2 _ kx2) 1/z Im [kz] -> O. (2) regrouping terms in the same powers of e, a one to

one correspondence can be obtained between the For a one dimensionally varying perfectly conduct- 0gn)s and the #(n)s. To the second order, the rela- ing surface the surface magnetic field satisfies the tionship is Helmholtz integral equation given by

24') = g(') (8)

H(r) = H{i)(r) + H(r') •n' G(r - r') ds' (3) 242) = g(2) + (g(1))2 (9) where G(r - r') is the free space Green's function. Hence in this paper we will limit ourselves to listing Here O/On denotes the normal derivative where the the f(n) coefficients, which are presented in the normal vector h = (5 - •hx)/(1 + hx 2) m, and h x is appendix, up to second order in e. Even though the the surface slope. Without loss of generality one two-dimensional scattering results are presented can rewrite this equation in terms of an unknown here, the same solutions can be deduced from the "source" functionf (x)defined by general three-dimensional scattering formulation

tk(i)x H(x) = 2 exp (' x )f(x). (4)

We use the boundary condition for the perfect electric conductor and obtain

[Rodr[guez, 1991]. The first class of theories we examine are the

so-called small perturbation method (SPM) theories [Rice, 1951; Winebrenner and Ishmaru, 1985a, hi. In this class of theories, the smallness parameter is, as usual, the product kzh - 0(e). Unlike horizontal

$(y) = • dx f(x) 1 + hx kx exp [ikzh ] exp [-i3•x] polarization we require h x - 0(e). The same re- (5) quirement is necessary for three-dimensional scat-

tering, regardless of polarization. Considering the where y = k x - k (i) is proportional to the field smallness of these two, we expect this type of .v x

momentum transfer in the x direction, 0 © is the expansion to be applicable to surfaces whose height incidence angle, and 15 denotes the Dirac delta is much smaller than the electromagnetic wave- function. For one-dimensional surfaces the equa- length and whose slope is •mall. We expect the tion plays an analogous role and is a consequence of SPM to provide better results when the incidence the extinction theorem in vector scattering. angle approaches grazing incidence. The first-order

The family of theories evaluated in this paper cross section predicted by SPM field perturbation is

KIM ET AL.: NUMERICAL SCATTERING FROM ROUGH SURFACES 517

the well-known Bragg cross section. The phase similar in flavor but different in details of implemen- perturbation version of SPM for vertical polariza- tation. We follow the approach of Rodrœguez [1989] tion has not been explored in detail. For vertical in this presentation. TSE differs from the previous polarization the SPM solution possesses resonance theories in that the surface is artificially subdivided terms which prevent extending perturbation solu- into a large-scale surface hr. (which is assumed to tions higher than second order [Nieto-Vesperinas, be smooth) and a small-scale surface hs (whose 1982]. For periodic surfaces this problem exists height is assumed to be much smaller than the even for the first-order solution due to the discrete electromagnetic wavelength). The perturbation ex- spectrum. Physically, the resonance occurs when pansion is carried in terms of two smallness param- the incident wave couples to the surface wave. eters, the large-scale surface slope and the product

In the second class of theories the momentum kzh s. The source function for the field perturbation transfer expansion (MTE) [Rodriguez, I989, 1991], is expanded as it is assumed that the ratio of the momentum

transfer in the x direction is small compared with the total field momentum; that is, •//k - O(e). We f(x) = D(x) f(n'm)(x) n!m----•- (10) expect this perturbation expansion to be valid n,m=o whenever the scattering surface is "smooth" (radius of curvature larger than the wavelength). The and the phase pertubation is obtained by analogy to well-known Kirchhoff (or physical optics) approxi- the previous methods. The likely domain of appli- mation can be shown to be given by the first-order cation for this theory is not clear, since it depends MTE. Since the Kirchhoff approximation is so on the two-scale splitting of the surface. In addition, widespread, we will evaluate it separately. the choice of the spectral split is not clear in this

It should be noted that in MTE, as in all the expansion. This issue will be addressed in detail in following perturbation expansions, we use D(x) = the next section. exp [-ik?)h(x)]. Brown [I982] has shown nonper- turbatively the correctness of this term for homo- 3. NUMERICAL IMPLEMENTATION AND RESULTS geneous rough surfaces. It has been shown by Rodriguez et aI. [this issue] and will be demon- In this section we just outline the numerical strated here that this phase factor is the key to the procedure for generating the simulated surfaces and success in the perturbation approach. calculating the scattered field, since the details are

The third theory, which we call the unified per- given in the horizontal polarization paper [Rod- turbation method (UPM), takes the smallness pa- rfguez et al., this issue]. We obtain the exact

--(i) rameter to be the product (to z - kz)h and D(x) = numerical solution for the surface current by using exp [-ikz(i)h(x)] We also assume that hx is of the the method of moments (MOM) [Harrington, 1968] same order as the smallness parameter. The small- for each surface in an ensemble. Approximate so- ness condition can be satisifed both in the small lutions are obtained by inserting the generated perturbation limit and in the small momentum trans- surface profile into the equations in the appendix. fer limit. In a separate paper, Rodriguez and Kim Our surfaces are periodic with period L, to avoid [1990] show that all the other approximations pre- edge effects. For periodic surfaces the far-field sented here can be derived as limiting cases of this modes for vertical polarization can be obtained approximation. It should be noted that the phase from perturbation version of this theory is identical to SPM phase perturbation; therefore we will only Hs(o )= Z Rmeikx'"+ikz•z (11) consider the field perturbation version. This ap- m proximation also suffers the same resonance problem as SPM. where the mode amplitude Rm is given by

The final class of theories considered are the

k two-scale expansion (TSE) theories [Brown, 1978; R,• = 2-•l•zm H(x')[cos Om McDaniel and Gorman, 1983; Rodriguez, 1989]. The term "two-scale scattering theories" has been applied to a varied group of theories which are -h x sin Om]e -tk .... '-ik•,(x') dx'. (12)


Here the mode wavenumber can be evaluated from TABLE 1. Summary of Surface Parameters

the grating equation as s o- h % Rc Lc Symbol

kxr n = k sin Om = kx © + m2•r/L. (13) -3 0.03 0.017 8.412 4.3 solid line -3 0.1 0.052 2.522 4.3 dashed line

In order to obtain the scattered field from a finite -3 0.2 0.104 1.260 4.3 short dashed line -3 0.4 0.209 0.630 4.3 dashed-dot line

surface we apply an illumination function to the -2.5 0.03 0.029 2.950 4.0 dashed line with circles incident wave using the method of Jordan andLang -2.5 0.1 0.098 0.884 4.0 solid line with crosses [1979]. From the far fields calculated from an en- -2.5 0.2 0.195 0.440 4.0 solid line with triangles

semble of surfaces we calculate both the backscat- The values of rms height (crh), rms slope (os), average radius tering cross section, of curvature (R c) (defined by Ulaby et al. [1986]), and correlation

length (L c) for the seven surfaces considered. The length units 2'rrr[(HsH*s) - (Hs)(Hs)*] are electromagnetic wavelengths. These values are related as e s

or0 = lim Left (14) = 0.522 o' h and R c = 0,252/o' h for the spectral slope of s = -3. r-->oo For the spectral slope of s = -2.5, % = 0.975 cr h and

R c = 0.088/cr h. where H s is the far field in the backscattering direction, and the bistatic scattering cross section,

2z'r[(HsH•)] A patch length of 25.6X is used since this is the •(0•) = lim ' (15) largest patch size which can be handled efficiently

Left with our computer. We use a point matching sam- r---• m

pling interval of 0.1A since consistent results are where Os denotes the scattering direction and Hs is obtained if the sampling interval is less than 0.1L the far field in this direction. The angle brackets To conform the accuracy of the numerical evalua- denote averaging over the ensemble of simulated tion of cr 0, the numerical results are compared with surfaces. Notice that cr 0 has the coherent field the analytic Bragg solution for the surfaces which removed, while tr is defined for the total field, rather satisfy the SPM condition. The numerical solutions than just the incoherent component. Here the effec- match, within the speckle noise, the Bragg predic- tive scattering area Lef t is defined as tion for the case where tr h is 0.03X. In addition,

Leer = f_• IXL(x)l 2 dx (16) where IL(x) is the illumination function [see Rod- dguez et al., this issue, Appendix B].

We select the surface spectral slope s to be -3

energy is conserved to better than 1%, on the average. The energy conservation condition is given by

• IRml 2 cos Om cos 0 (i) m

= 1. (17)

and -2.5. These spectral slopes are similar to those These tests fissure the accuracy of the numerical observed for the ocean surface heights [Phillips, solutions. 1980]. Note that these surfaces are the one-dimen- The MOM backscattering cross sections for the sional analogues of two-dimensional surfaces with surfaces listed in Table 1 are shown in Figure la. s = -4 and s = -3.5 spectral slopes. Spectral For slightly rough surfaces (tr h = 0.03, 0.1A), tr0 slopes ors = -2.5 can also be found in some natural shows the Bragg cross section. Notice that the soil surfaces. Table 1 summarizes the surface pa- backscattering cross section remains almost con- rameters used for evaluating different scattering stant as the incidence angle approaches 60 ø. The theories considered in this study. surfaces with spectral slope of s = -2.5 are brighter

Several related parameters used for the numerical than the ones with s = -3. The error bars in Figure scattering calculation are the number of surfaces to l a are approximately _+ 1 dB, which is slightly larger be averaged, the surface period, and the point than the size of the symbols used in Figure la. matching sampling interval. Fifty surfaces were Unlike the horizontal polarization case we are not used for the ensemble average. For this number of able to simulate the scattering from rough surfaces surfaces the speckle noise for a 90% confidence with large rms slope (o s > 0.2), since the power level is estimated to be _ 1 dB from the mean value. conservation error of these surfaces is, on the


• .10, o

• -20' • .

o

MOM

0 ' 1'0 ' 2'0' ' 3'0 ' 4'0 ' 5'0 ' 6 0 INCIDENCE ANGLE (DEGREE) a

-1 .o .o.5 o.o o.5 1 .o

SIN(theta) b

.0 -0.5 0.0 0.5 1.0

$1N(thata) C

10

o

-lO.

:

.2o-

ß

olo oi. •.o SIN(theta) d

average, larger than 1%. Hence the comparison of the approximations does not explore as large a regime of validity as was possible for horizontal polarization. However, the surfaces considered here are rough enough to show the limitations of most theories.

The MOM bistatic cross sections for the same set

of surfaces and for incidence angles of 0 ø, 25 ø, and 50 ø are presented in Figures lb, lc, and ld, respectively. As can be seen from these figures, as the surface rms height increases, the surfaces with large rms height (•r h = 0.4,•) scatter more isotropically. For lower rms heights the specular component can be clearly distinguished. The scattering patterns with similar rms slope resemble each other.

We used the results shown in Figure 1 as a yardstick to measure the accuracy of the scattering cross sections predicted by the approximate theories examined here. In Figures 2-8 that follow we present the difference in decibels between the MOM scattering cross section and the cross sections predicted by each approximation. A positive value for the error means that the MOM results are

brighter than the prediction of the approximate theory. In the following analysis a theory will be called accurate in a given regime if the magnitude of the cross-section error is less than 1 dB.

Small perturbation method ($PM). The first- and second-order solutions of both the field and phase expansion of SPM are examined. From the SPM currents shown in the appendix it is clear that the current may diverge due to a singularity if kz(t + k•) becomes zero. For aperiodic surfaces the singularity is integrable, up to the second order. For periodic surfaces the resonance exists for several discrete angles even in the first order solution since the surface spectrum becomes discrete. The condition for this resonance is that the discrete surface spectrum number n satisfies n = L/)t (+-1 - sin O(i)). Here we made the comparisons for nonresonant incidence angles. However, the scattering cross section of SPM becomes inaccurate, especially for the phase expansion, when the incidence angle is close to the resonance angles. This problem also exists for the unified perturbation method which will be discussed later in this section. Hence the

Fig. 1. (Opposite) (a) Method of moments backscattering cross section and bistatic cross sections for (b) 0 ø, (c) 25 ø, and (d) 50 ø incidence angles. Table 1 contains the symbols correspond- ing to each surface type in Figures 1--8.


perturbation can be done up to the second order for aperiodic surfaces if SPM or UPM is used for vertical polarization. For periodic surfaces the resonance exists for any order. The results for the first-order SPM field expansion are shown in Figure

2 and the second order in Figure 7. As one can see from Figure 2a and Figure 7a, the second-order solution improves the accuracy for the coherent component. However, the second order is worse than the first order as the rms height of the surface becomes large. As expected from the smallness parameters (or h and rr s) used in this expansion, the method predicts erroneous scattering cross sections if the surface becomes rougher; in this case, SPM vastly overestimates the surface brightness. As shown in Figure 2 a, these errors decrease for larger incidence angles. This, too, is to be expected since the smallness parameter is ko' h cos 0 (i). It can be noticed, also, that the convergence of the method is not affected by the rms slope or curvature of the surface. This can be deduced from the similar behavior of the scattering cross section for two different surface spectra of the same rms height but different rms slopes and curvatures. Even though a small rms slope has been assumed for this method, the solution appears to be unaffected by the slope.

From the bistatic cross sections shown in Figures 2b-2d one sees that the first-order SPM field expansion produces erroneous results for large rms height and near the specular direction. This method works for surfaces with rms height less than 0.1X. The validity conditions usually associated with

first-order SPM are [Ulaby et aI., 1986] kcr h < 0.3 (18) rrs < 0.3 (19)

Recently, Chen and Fung [1988] have shown numerically the need for a small correlation length when the surface spectrum is Gaussian:

kLc < 3.0. (20)

We note that for the power spectra considered here the height requirement seems too stringent by at least a factor of 2, while the slope requirement has

Fig. 2. (Opposite) (a) Difference between field perturbation SPM and MOM for the backscattering cross section and the bistatic cross sections for (b) 0 ø, (c) 25 ø, and (d) 50 ø incidence angles.

SPM FIELD (FIRST ORDER)

3

I L -m,l

INCIDENCE ANGLE (DEGREE)

0.0

SIN(theta) 'b 0.5 1 .o

.o .o'.s olo 0.5 C 4.0 SiN(theta)

• 5

o

• -10 .... ß ß - .

SIN(theta) d -0'.5 1.0


5

SPM PHASE (FIRST ORDER)

0 10 20 30 40 50 60

INCIDENCE ANGLE (DEGREE) a

•. 5 • o DEGREES

m -1.0 -0.5 0.0 0:5 1.0 SIN(theta) b

• 5

O 1

m -1,0

__

ß

-o.5 o.o o'.5 1 .o SIN(theta) C

• :I 50 DEGREES • - o

.1 .o -o.5 o.o o.$ 1 .o SIN(theta) d

not been tested. For power law surfaces there does not seem to be any requirement for a correlation length.

The phase expansion of SPM produces very accurate results for surfaces with modest rms slope. For the surfaces with larger rms slope the solution diverges for incidence angles close to the resonance incidence angles. (see Figure 3a). As can be seen in Figures 3b-3d, the bistatic cross sections are accurate except for 50 ø incidence angle which is very close to a resonance angle. Notice that the bistatic cross section oscillates wildly near the resonance angles. The second-order solution (see Figure 7b) is worse than the first-order solution near the resonance angles.

Except near the resonance angles the phase expansion produces a better solution than the field expansion. It appears that the field SPM is less affected by the resonance than the phase perturbation. Considering overall performance, the phase SPM is better than the field SPM.

Kirchhoff. Since Kirchhoff is the first-order approximation of the momentum transfer expansion, % predicted by the Kirchhoff approximation is expected to be accurate for smaller incidence angles. As shown in Figure 4a, this appears to be true up to 20 ø incidence angle. However, the errors increase sharply as the incidence angle increases. It is interesting to notice that both back and bistatic scattering errors behave in the same manner regardless of the roughness. The bistatic error increases monotonically as the momentum transfer gets larger.

The regime of validity usually quoted in the literature for the Kirchhoff approximation is [ Ulaby et al., 1986]

Rc ;> • (21)

kLc > 6. (22)

The second condition has not been tested by our examples, but the first condition seems to be too restrictive for the power law surfaces considered here. For a Gaussian roughness spectrum the validity of the Kirchhoff approximation is also given by

Fig. 3. (Opposite) (a) SPM phase first-order perturbation cross-section errors for the backscattering cross section and the bistatic cross sections for (b) 0 ø, (c) 25% and (d) 50 ø incidence angles.

• 3

,n

........

1 0 2'0 3 0 4'0 5'0 6 0 INCIDENCE ANGLE (DEGREE)

• 5 co

.1 .o -o.5 o.o 0.5 b 1.o SIN(theta)

•, :1 MTE PHASE

0 10 20 30 40 50 60

INCIDENCE ANGLE (DEGREE) •

SIN(theta)

0 DEGREES

•--- -,____.•. _-•

25 DEGREES

.0 -0.5 0,0 0.5 1.0

SIN(theta) C

-1.0 -0.5 0.0 0.5 1.0

SIN(theta) d

• 25 DEGREES

.1.0 -0.5 0.0 0.5 1.0

SIN(theta) C

" '".,.'o ' .o'., ' o;o ' oi, ' SIN(theta) d

50 DEGREES

1.0

Fig. 4. (a) Kirchhoff cross-section errors for the backscatter- Fig. 5. (a) MTE field perturbation cross-section errors for the ing cross section and the bistatic cross sections for (b) 0 ø, (c) 25% backscattering cross section and the bistatic cross sections for and (d) 50 ø incidence angles. (b) 0% (c) 25 ø, and (d) 50 ø incidence angles.


Thorsos [1988]. However, on the basis of our examples, a major determinant of the regime of validity of the Kirchhoff approximation is the incidence angle, regardless of rms height.

Momentum transfer expansion (MTE). We considered the field and the phase perturbations of the second-order momentum transfer expansion. Since the phase expansion of MTE is better than the field expansion for large incidence angles as the surface becomes rougher, we show (Figure 5) only the results of the phase expansion of the MTE. The backscattering cross section of the phase MTE is accurate if the incidence angle is smaller than 30 ø as predicted by Rodrfguez [1991]. However, rr 0 of the phase MTE diverges as the incidence angle becomes larger than 30 ø . This divergence is much slower than for the horizontal polarization case. This second-order solution provides better accuracy than Kirchhoff (the first-order MTE solution) for small incidence angles (<30 ø ) because of the effective averaging of the surface slopes. Unlike horizontal polarization the backscattering cross section is not exact even near the nadir direction for

the surface with large rms slope (a s = 0.2). This approximation overestimates the backscattering cross section while the Kirchhoff approximation underestimates the cross section. As the rms slope becomes larger, it is interesting to notice that the backscattering cross section is more accurate for large incidence angles. Examination of the bistatic cross section (Figures 5b-5d) shows that the error decreases in the regions of small momentum transfer; that is, close to the specular direction. How- ever, if the momentum transfer is large, the solution may be worse than the Kirchhoff prediction. Con- sidering Figure 5a, it appears that the validity of this method is strongly dependent on the incidence angle.

Unified perturbation method (UPM). The field expansion of UPM is considered here since the phase expansion of this method becomes identical to the phase expansion of SPM which was considered above. As discussed in SPM, this method also suffers the resonance problem for periodic surfaces. We consider incidence angles which are not resonance angles. Considering both the,first- and sec-

\.

Fig. 6. (Opposite) (a) UPM field first-ordef,,perturbation cross-section errors for the backscattering cross sedtion and the bistatic cross sections for (b) 0 ø, (c) 25 ø, and (d) 50 • incidence angles.

UPM FIELD (FIRST ORDER)


-1.0 -0.5 0.0 0.5 1.0

SIN(theta) b

.1,o -0.5 0.0 0.5 1.0

SIN(theta) C

, .

-1.0 -0.5 0.0 0.5 d 1.0 SIN(theta)


• SPM FIELD (SECOND ORDER)

0 10 20 30 40 50 60

INCIDENCE ANGLE (DEGREE) a

5

SPM PHASE (SECOND ORDER)

• 3

....... ......... .... • -1 • / :

0 10 2o 30 4o 5o 6o

INCIDENCE ANGLE (DEGREE) b

UPM FIELD (SECOND ORDER)

'\ /\/

10 20 30 40 50 60 c


a resonance angle. Even for this angle, this approxi. mation is much better than other SPM approxima. tions. This field expansion of UPM seems to be the best method among the approximations considered in this paper if the resonance angles are avoided. As mentioned in SPM, this singularity' comes from the peridioc surface assumption. The integral evaluation of the analytic expression of this theory can avoid the resonant behavior of the theory up to the second order. This is currently under investigation. This method provides very accurate results since the phase factor D relaxes the strict small height requirement of the SPM field perturbation.

Two-scale expansion (TSE). The spectrum of the rough surface is split into two parts. One corresponds to the small surface component, and the other corresponds to the large but smooth surface component. Both phase and field expansions of TSE were considered. The phase expansion of TSE is almost identical to the field counter-

part. Hence we present here only the field expansion of TSE. The first-order solution is imple- mented such that the first-order UPM field appears as a limit since it seems to be the most accurate

method, except near the resonant incidence angles. A critical factor in the evaluation of the TSE is

the selection of the spectral split. Various criteria have been proposed for selecting this parameter, Brown, 1978; Durden and ¾esecky, 1990]. We use a conventional spectral split, which is close to the incidence wavelength. Specifically, from the nadir to 15 ø, the spectral split wavelength is chosen to be 0.53. since the Kirchhoff approximation is accurate here. From 20ø-40 ø incidence angle the spectral split wavelength is 13.. Otherwise, we use 23. as the spectral split wavelength. There is no particular trend in the scattering behavior caused by the

Fig. 7. (a) The second-order backscattering errors for the SPM arbitrariness of the spectral split. The backscatter- field (b) SPM phase and (c) UPM field. ing cross-section error is less than 1.5 dB for the

surfaces considered here (see Figure 8a). When both the Kirchhoff approximation and the first-

ond-order solutions, it is found that the first order order UPM field expansion produce accurate re- appears to be more accurate than the second order, suits, the TSE is insensitive to the spectral split. especially near the resonant angles (see Figure 6a The backscattering cross section of TSE can be and Figure 7c). As shown in Figure 6a, the method more accurate than that of UPM field near the predicts accurate backscattering cross sections for resonant incidence angles since the small surface almost all cases considered here. The backscattering components can be chosen to exclude the resonant error is always less than 1 dB. As the rms slope of the surface component. The bistatic .cross section be- surface increases, the method degrades gracefully. haves similarly to the Kirchhoff case when the The bistatic cross sections are also extremely accu- spectral split wavelength is 0.53.. When the split rate, except for 50 ø incidence angle, which is close to wavelength becomes larger, the bistatic cross sec-


5'

m

-5 0

TWO SCALE FIELD (FIRST ORDER)

10 20 30 40 50 60


-1.0 .0.5 0.0 0.5 1.0

SIN(theta) b

25 DEGREES

' o•o ' o•s C4'ø SIN(theta)

• 5 t 50 DEGREES

-1,0 -0.5 0.0 0.5 d .0 SlN(thete)

tion becomes a hybrid of the Kirchhoff and UPM field. A more extensive comparison of other TSE theories, together with the appropriate spectral split, will be presented elsewhere [Kirn and Rod- rfguez, 1992].

4. CONCLUSIONS

We have presented a comparison of the bistatic and backscattering cross sections predicted by a family of analytical scattering theories with numerical results obtained by using the method of moments. The following list summarizes our conclusions based on the results presented above.

I. Unlike for horizontal polarization, SPM re- quires the surface slope to be small. In addition, the solution exhibits resonant behavior for several inci-

dence angles, if periodic surfaces are considered. For aperiodic surfaces, which possess a continuous spectrum, this resonance prohibits this perturbation to proceed higher than the second order. The first- order solution is better than the second order,

except for the coherent component. The field SPM method works well as long as the rms surface roughness does not exceed one-tenth of the electromagnetic wavelength. The theory overestimates the scattering cross section rather badly for rms heights which are significantly larger than this value. As expected, the divergence tends to decrease with incidence angle since the smallness parameter also decreases with incidence angle. The surface rms height is a dominant factor in the accuracy of the results.

2. The first-order phase perturbation method works well in most cases for both the bistatic and backscattering cross sections. It tends to diverge as the incidence angle approaches the resonant angles. The divergence is even worse for the second-order solution.

3. The Kirchhoff approximation diverges monotonically as a function of incidence angle. It is an adequate approximation for the smoother surfaces and for incidence angles smaller than 20 ø . In con- trast with SPM, the errors are almost independent of the rms heights, rms slopes, and rms curvatures considered here. For both horizontal and vertical

Fig. 8. (Opposite) (a) TSE first-order cross-section errors for the backscattering cross section and the bistatic cross sections for (b) 0 ø, (c) 25 ø, and (d) 50 ø incidence angles. The spectral splits for various incidence angles are given in the text.


polarizations the accuracy of this approximation is Small perturbation method strongly dependent on the incidence angle.

4. The second-order MTE expansion is good as D(x) = 1 (24) long as the momentum transfer is small. Specifi- (0) cally, the theory works very well in the backscat- fs•M(x) = 1 (25) tering direction if the incidence angle is smaller than -i k(i) ) H(t)eitX approximately 30 ø. Considering both horizontal and f•M(x) = • kz(t + x + kz(t + k?)) dt vertical polarizations, the phase expansion is better than the field expansion. (26)

5. The best results were obtained by using the 1 { UPM even though the comparisons are not satisfac- f•M(X ) = (-•)2 f_• f-•o• kz2(t• + k(xi)) - 2kz(t, + k (i)) tory due to the resonances introduced by periodic • surfaces. For both the bistatic and backscattering cross sections this theory presented errors which were on average smaller than the ones obtained by using other theories. The theory seems to work least well when the surface height is very large. In order to remove the effects of the resonance the analytic expansion of UPM should be evalauted to compare with MOM. This is currently under investigation.

6. The version of TSE which we studied had the

(t2 + kx(i))t21 ß kz(k} i) + t2) + kz(t2 '• kx(i))' ] + 2(t, + kx(i))t2

(tl+ kx (i)) (t2 + kx(i))t2 ]} - 2 kz(tl + kx(i)) (tl - t2) /cz(kx © + t2) +kz(t 2 + kx(t))tl ]j

H(tl - t2)H(t2)e ittx dt2 dt• (27)

first-order UPM and Kirchhoff as its limits. The con- where kz(x) are defined in (2). ventional spectral split produces accurate scattering cross sections. For vertical polarization this approach Momentum transfer expansion has advantages over UPM since the resonance can be avoided by properly choosing the spectral split. How- D(x) = exp [-ikz(i)h(x)] (28) ever, overall optimization can be obtained from the f•.(x) = 1 (29) first-order UPM as one of the TSE limits. More

extensive comparisons between UPM and TSE will i fMTE(X) = 1 + • hxx (30) be presented elsewhere [Kim and Rodrœguez, 1991]. (2)

As a final remark we note that the assumption 2k cos 3 (0 ©) made in this study, which we can only justify where a subscript xx denotes the second partial heuristically, is that the theory will be applicable to derivative with respect to x. two dimensionally varying surfaces with similar characteristics if it works for both polarizations for Unified perturbation method one dimensionally varying surfaces. Further work remains to be done on comparing the three-dimen- D(x) = exp [-ikz(i)h(x)] (31) sional scattering predictions of the theories pre- (0) sented here, including the predicted depolarization, fuw(x) = 1 (32) with numerical and experimental results.

-i/_•o• ([ kz(i) /•M(x) = • kz(t + •')) - APPENDIX

In this appendix we list the source funcntions (t + kx(i))t •\ 't evaluated in this paper to second order in the +kz(t + kx(i))J)H(t)etX dt smallness parameter with the exception of the two-

scale expansion. We will denote the Fourier transform of the surface height by H(t), which is given by f•)•u(x) = (-•)2 [kz(t• + kx (i)) --/•/)]2 H(t) = •_• h(x)e -itx dx. (23) _ 2[kz(t + k(xi)) _ k•i) ]

(33)


(tl + kx(i))t2 (tl + kx (i)) + 2[kz(t 1 + kx (0) - k•i) l kz(tl + kx(i)-•- • - 2 kz(t, + kx(i)•

ß (,,- ,•) •(•1 ') + ,•)- •7 ) + •z(,2 + H(tl - t2)H(t2)e it•x dr2 dtl. (34)

Two-scale expansion

D(x) = exp [-ik•Oh(x)] (35)

f (•,0)=f•EIh=h; (36) TSE

f(0•) ½0) (37) T•E :JUPM h = hs

where h• •d h s are the large and small components of the surface, respectively.

Acknowledgments. The rese•ch described in t•s paper was

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Y. Kim, E. Rodriguez, and S. L. Durden, Jet Propulsion Laboratory, 300-319, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109.