a comparison of back scattering

8/6/2019 A Comparison of Back Scattering

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 33, NO. 1, JANUARY 1995 195

A Comparison of BackscatteringModels for Rough SurfacesK. S . Chen, Member, IEEE, and Adrian K. Fung, Fellow, IEEE

Abstract-The objective of this study is to examine the ease ofapplicability of three scattering models. This is done by consid-ering the time taken to numerically evaluate these models andcomparing their predictions as a function of surface roughness,frequency, incident angle and polarization with the momentmethod solution in two dimensions. In addition, the complexityof the analytic models in three dimensions and their analyticreduction to high and low frequency regions are also compared.The selected models are an integral equation model (IEM), a fullwave model (FWM), and the phase perturbation model (PPM).It is noted that in three dimensions, the full-wave model requiresan evaluation of a 10-fold integral, the phase perturbation modelrequires a 4- and 2-fold integral while the integral equationmodel is an algebraic equation in like polarization under singlescattering conditions. In examining frequency dependence ofIEM and PPM in two dimensions numerically, the same modelexpression is used for all frequency calculations. It is found thatboth the IEM and PPM agree with the moment method solutionfrom low to high frequencies numerically. The FWM given in[lo] agrees only with the Kirchhoff model.

I. INTRODUCTIONN the development of a theoretical model for wave scat-I ering from randomly rough surfaces much effort has been

devoted to broadening the ranges of validity over the classicalKirchhoff and small perturbation models and to bridgingthe gap between these two models. Three models that haveso broadened their ranges of validity are selected for com-parison with the moment method solution. They are thephase perturbation model (PPM) [l], [ 2 ] ,a full wave model(FWM) [3]-[6], and an integral equation model (IEM) [7],[8]. The comparisons are aimed at the case of applicationof these models by examining their numerical evaluationand mathematical complexity. In Section 11, the three modelexpressions in three dimensions are summarized and theirreductions in the high and low frequency regions are examined.Due to the complexity of some of these models only thetwo-dimensional versions of these models are numericallyevaluated in Section 111. The relative amount of time taken todo these numerical computations and the associated stabilityproblems are discussed. Conclusions are drawn in Section IV .

Manuscript received June 25, 1993; revised June 9, 1994. This work wassupported by the National Science Council of Taiwan.K. S . Chen is with the Center for Space and Remote Sensing Research,National Central University, Chung-Li, Taiwan, Republic of China.

A. K. Fung is with the Wave Scattering Research Center, Department ofElectrical Engineering, University of Texas at Arlington, Arlington, TX 76019USA.IEEE Log Number 9406430.0196-2892/95$O4

11. SURFACE ODELSThis section gives the three-dimensional mathematical ex-

pressions for the selected models. The complexity of theseexpressions and how they reduce analytically to known modelsare discussed.A . Phase Perturbation Model

From [1, (31)], the backscattering coefficient is expressed as

where -

P ( K ; ) = k , ( K i ) / k . ( 5 )W(*;) s the normalized spectrum of the surface roughnesssuch that-In the equations above, K;= (kiz, iz) s the incident wave

vector, IC is wavenumber, and R d = R I-R2 is position vectorrepresenting distance between two points on the surface.The PPM is originally derived for scalar waves. In Wine-brenners dissertation, only the HH polarization case wasderived. Hence, it does not include polarization dependence.Calculations of the backscattering coefficient using (1) in-volves a 4-fold integral and a 2-fold integral.Next, consider the reduction to known models. In the lowfrequency limits where ka is small, one can expand theexponential in (1) in a power series and keep up to secondorder terms. Equation (1) can be shown to reduce to thesmall perturbation model in horizontal polarization [11. In

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196 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 3 3 , NO. 1, JANUARY 1995

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-4- PMx EWM ( re f 10)A M M (HH)

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0 10 20 30 40 50 60 70IncidenceAngle (deg)

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S EM (HH)- SPM (HH)1,?:;,(L;, , , , , , I , , , , I \ , , , , , , , , ,

MM (HHI

0 IO 20 30 40 50 60 70Incidence Angle (deg)

(c )

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- PM0 IO 20 30 40 50 60 70

Incidence Angle (deg)(b)

A EM (HHI- - SPM (HH)- PMx F ~ M re f 10 )

A MM (HH)

0 IO 20 30 40 50 60 70Incidence Angle (deg)

(d )Fig. 1 .(c ) k a = 0.6, (d) k o = 0.8,Model predictions of roughness behavior as compared with exact moment method simulation ( k L = 4.188); (a) ko = 0.2, (b) k o = 0 .4 ,

the high frequency limit, according to [l] the PPM differsfrom the Kirchhoff model by a factor of cos48. However, in[2] it is shown that PPM reduces numerically to KM for thetwo-dimensional problem. According to [2], the discrepancybetween the analytical and numerical results is due to thenecessity of retaining higher order terms in the expansion ofthe p function. These terms are responsible for the reductionof PPM to KM .B. Full-Wave Model

The full-wave model given in [5 , (9)] is

where subscripts 1 and 2 denote two points on the surface, andp , q denote the polarization of the scattered and incident waves,respectively.A , is the projected area on the reference plane.Bahar commented in [5] that "the numerical evaluation of the10-fold integral is, in general, too time-consuming for practicalpurpose." In this paper he also stated that "in order to considerproblems that are numerically tractable (using minicomputers)only one-dimensionally rough surfaces are considered." Away to simplify (7) is to assume that the su rjk e slopes areindependent of the heights [5] and

Under these assumptions (7 ) can be reduced to two doubleintegrals if the characteristics functions of the rough surfaceheight have analytic forms.The reduction of the full-wave solution in the low frequencyregion has been discussed in [lo] where it is indicated thatBahar's analytic argument for the reduction of the full-wavemodel to the perturbation model is based on an inconsistentexpansion in small surface height. They further reported that"for backscattering with shadowing neglected, the full-wave


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197CHEN AND FUNG A COMPARISON OF BACKSCATTERING MODELS FOR ROUGH SURFACES

0

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-30

-40

10

-40

--8- EM I H H I

- PM0 10 20 30 40 50 60 70

Incidence Angle (deg)(a)

A - -j - . - . :. i 1j: - - - :.................. .................. ................. . . . . . .. - 'c - - i: 4._...........I : .-- : - - 4......... .................................. : . : .. : .:.j 0 1 ~%...... -1EM (W l ...... ................... j .................... ?..:.-...._.....i ................Y2'C - KM_ SPM (W)

-30 1 ...... .... C-^^..^ ICY Behavior ofModels ........i as Functionof ncidence Anglej (kL= 3.14, ko = 1.0,Wpol.)..... - .....................................................0 10 20 30 40 50 60 70

Incidence Angle (deg)(b)

Fig. 3. Model predictions of frequency behavior as compared with exactmoment method simulation (ICL =3.14, ku =1.0, (a) HH polarization, (b)VV polarization.

or

x w(") ( Icin o - U, )dudw (12)where 0 is the incidence angle and

I , = (ka os0 ) n [ 2 n e - ~ ~ ~ ~ s in201 (13)and the plus sign is for ww polarization while the minus signis for hh polarization.

(10) is the Fourier transform of the nth power of the surfacecorrelation function p. Equation (10) only accounts for single


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CHEN AND FUNG: A COMPARISON OF BACKSCATTERING MODELS FOR ROUGH SURFACES 199

TABLE ISAMPLES OF CPU TIMEREQUIRED BY MODELS(USING CONVEXUPERMINI-COMPUTER)

FWMIEM 1.25 0.84 0.95

varying between 0.2 to 0.8 in steps of 0.2. We choose thesesets of parameters because they lie between the regions ofvalidity for the KM and the SPM. Note that only horizontalpolarization is considered here. As can be seen in the figures,IEM and PPM accurately predict the scattering curve over allangles, while the FWM ives a result in agreement with theKh t As lca increases, FWM is closer to MM because thesurface is closer to a Kirchhoff surface when L is comparableto wavelength. Discussions on the range of validity can befound in [9]. It should be emphasized that the FWM usedhere is from [lo]. Figs. 2 4 present the frequency behaviorof the backscattering coefficient as frequency changes fromkL =1.57,ka = O S to lcL =6.28,ka =2.0 in two steps eachtime by a factor of 2. Part (a) of each figure is for horizontalpolarization and part (b) is for vertical polarization. In Fig. 2(a)the surface parameters are larger than those appropriate forthe SPM. The IEM and PPM closely follow the MM resultsbut FWM s on the high side. Fig. 2(b) shows a comparisonbetween IEM and MM for vertical polarization to demonstratethe polarization dependence of E M. Similar illustrations areplotted in Fig. 3 where k L =3.14 and ko =1.0. In this case,we are in the intermediate frequency region. As we can seefrom the figure, both IEM and PPM continue to match theMM results for horizontal polarization and FWM also getscloser to MM and is in close agreement with the Kirchhoffmodel. Results based on a further increase in frequency tok L =6.28 and lca =2.0 are indicated in Fig. 4. Here, IEM,PPM, KM and FWM re all in agreement with the results ofMM.

To test computational efficiency we use the cases inFigs. l(a), 2(a), and 4(a). All calculations are performed onCONVEX supermini-computer using double precision. Theevaluation of the integrals in PPM and FWM are carried outusing the Filon algorithm [121 which is suitable for cosine-sineintegrals. First consider the case of Fig. l(a) where IEM takes1.25 s CPU time, PPM 33.4 s, and FWM 2.11 s to produce.To obtain the convergence results in Fig. 2(a), IEM takes 0.84s CPU time, PPM 5.04 s, and FWM 1.07 s. When the surfacebecomes rougher, as shown in Fig. 4(a), E M takes 0.95 s,PPM 112.9 s, and FWM 2.06 s. It is obvious that as thesurface becomes rougher (Figs. 2(a) and 4(a)) or the surfaceslopes becomes smaller (Fig. 1 a)), the computational timerequired by PPM increases significantly as compared to IEMand FWM. It should be noted at this point that by introducingappropriate asymptotic expansions, simplifying the evaluationof the PPM in the high roughness limit are reported in [131.Table I summarizes the computational efficiency. Finally, wesummarize our observations in Table I1 to indicate the majorproperties of the models.

TABLE IISUMMARY OF MODELCOMPARISONS

D=disagnewith MM; = a p with MM.

IV. CONCLUDINGEMARKSComparisons of three surface models for perfectly con-ducting surfaces and HH polarizations are made from thestandpoint of mathematical complexity and numerical com-putations. Among the three models the most complex isthe FWM. From a computational standpoint the simplestmodel is the E M . The PPM is not fully polarimetric at thispoint. Comparisons for finitely conducting surfaces and otherpolarizations should be of interest for future investigations.

ACKNOWLEDGMENTThe authors are indebted to the anonymous reviewers fortheir penetrating stimulating comments that substantially in-fluenced this final presentation.

REFERENCES[I ] D. P. Winebrenner and A. Ishimaru, Application of the phase-

perturbation technique to randomly rough surfaces, J. Opt. Soc. Am ,A., vol. 2, no. 12, pp. 2285-2293, 1985.[2] S . L. Broschat, L. Tsang, A. Ishimaru, and E. I. Thorsos, A numericalcomparison of the phase perturbation technique with the cla ssical fieldperturbation and Kirchhoff approximations for random rough surfacescattering, J . Electro Waves Applic., vol. 2, no. 1, pp. 85-102, 1987.[3] E. Bahar, Full-wave solutions for the scat tered radiation fields fromrough surfaces with arbitrary slope and frequency, IEEE Trans. Ant.Propagat., vol. AP-28, pp. 11-21, 1980.

[4] -, Scatteringcross sections for random rough surfaces: Full-waveanalysis, Radio Sci., vol. 16, pp. 331-341, 1981.[5] -, Full-wave analysis for rough surface diffuse, incoherent radarcross sections with height-slope correlations ncluded, IEEE Trans. Ant.Propagat., vol. 39, pp. 1293-1304, 1991.[6] E. Bahar and Y. F. Li, Scattering cross sections for non-Gaussiansurfaces: Unified full wave approach, IEEE Trans. Ant. Propagat., vol.

[7] A. K. Fung and G. W. Pan, A scattering model for perfectly conductingrandom surfaces, I. Model development, Int. J. Remote Sens. vol. 8,no. 11, pp. 1579-1593, 1987.[8] A. K. Fung, Z. Li, and K. S . Chen, Backscattering from a randomlyrough dielectric surface, IEEE Trans. Geosci. Remote Sensing, vol. 30,pp. 35&369, 1992.[9] E. I. Thorsos, the validity of the Kirchhoff approximation for roughsurface scattering using a Gaussian roughness spectrum, J. Acoust. Soc.Am., vol. 83, pp. 78-92, 1988.[IO] E. I. Thorsos and D. Winebrenner, An examination of the full-wavemethod for rough surfaces scattering in the case of small roughness,J.Geophys. Res., vol. 96, no. c9, pp. 17107-17121, 1991.[ I l l E. Bahar, Examination of full wave solutions and exact numericalresults for one-dimensional slightly rough surfaces, J. Geophys. Res.,vol. 96, no. c9 , pp. 17123-17131, 1991.[121 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions.New York: Dover, 1965.[I31 K. Ivanova, M. A. Michalev, and 0. I. Yordanov, Study of the phaseperturbation technique for scattering of waves by rough surfaces atintermediate and large values of the roughness parameter, J. Electro.Waves Applic., vol. 4, no. 5, pp. 401414, 1990.

39, pp. 1777-1781, 1991.


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200 IEEE TRANSACTIONS ON GEOSCIENCE AN D REMOTE ENSING, VOL. 33,NO. 1, JANUARY 1995

K. S. Chen (M94) graduated from National TaipeiInstitute of Technology, Taipei, Taiwan and receivedthe Ph.D. degree in electrical engineering from theUniversity of Texas at Arlington in 1990.From 1985 to 1990, he was with Wave Sca t-tering Research Center at the University of Texasat Arlington. He is now an Associate Professor atthe Center for Space and Remote Sensing Research,National Central University, Taiwan. His major re-search has been in the areas of wave scattering fromterrains and sea, radar signal and image simulationand analysis, and their applications to remote sensing.

Adrian K. Fung (S6(rM6&SM7(rF85) wasbom December 25, 1936, in Liuchow, Kwangsi,China. He received the B.S.E.E. degree from TaiwanProvincipal Cheng Kung University, the M.S.E.E.degree from Brown University, Providence, RI, andthe Ph.D. degree from the University of Kansas,Lawrence, in 1958, 1961, and 1965, respectively.He was a faculty member in the Electrical En-gineering Department of the University of Kansasfrom 1965-1984, becoming a full Professor in 1972.He has been a Professor of Electr ical Engineering atthe University of Texas at Arlington since 1984. He is currently the Directorof the Wave Scattering Research Center. His research interests include wave

scattering and emission from irregular surfaces and random media, radar imagesimulation, numerical simulation of radar scattering, and radome analysis.He is the author of Microwave Scattering and Emission Models and TheirApplicarions and co-author of a three-volume book on microwave remotesensing.Dr. Fung is a recipient of the 1987 Halliburton Excellence in ResearchAward, the 1989 Distinguished Research Award from the University of Texasat Arlington, and the 1989 Distinguished Achievement Award from the IEEEGeoscience and Remote Sensing Society. He is a member of Sigma Xi andU.S. Commission F of the Intemat ional Scientific Radio Union.

a comparison of back scattering

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