A Comparison Study of the Surface Scattering Models and Numerical ModelH.T. Ewe1, Joel T. Johnson2 and K.S. Chen31Faculty of Engineering, Multimedia University,Cyberjaya 63100, Selangor, MALAYSIAEmail: [email protected] of Electrical Engineering and Electro-Science Laboratory,The Ohio State University, Columbus, OH 43210 USA3Center for Space and Remote Sensing Research and Institute of Space Science,National Central University, Chung-Li, Taiwan 32054Abstract – This paper describes a comparison study of surfacescattering models and a numerical model for dielectric surfaces.Two surface scattering models namely the Integral Equation Model(IEM) and the Small Slope Approximation (SSA) model are used tocalculate the backscattering coefficients of rough surfaces and theresults are compared with numerical simulations based on theMoment Method (MoM). Analysis of the results obtained is alsopresented.I. INTRODUCTION In the study of the interaction of microwaves with mediabounded by rough interfaces, a practical and realistic surfacescattering model is required to understand the surfacescattering mechanism and predict the scattering returns for avariety of surface profiles. The usability and correctness ofthe model are essential in microwave remote sensing and agood understanding of the validity and limitation of themodels will be necessary for modeling active and passivemicrowave scattering mechanisms for earth terrain and alsoinversion models. Surface scattering models based on boththe small perturbation method (SPM) and the Kirchhoffmodel had been widely used in the past in the theoreticalmodeling of microwave remote sensing [1]. Recent modelssuch as the Integral Equation Model (IEM) [2] and the SmallSlope Approximation (SSA) model [3] have shown greatpromise in the prediction of surface scattering returns fromvarious research studies. In this paper, a comparison study ofthe IEM model and the SSA model is carried out fordielectric rough surfaces. To compare the predictions ofbackscattering coefficients from these two models with theactual backscattering returns from the surface, numericalsimulations based on the Moment Method (MoM) aregenerated for the comparison. It is generally found that boththe IEM and the SSA model with the inclusion of up tosecond order surface slope are good models in comparisonwith the predictions from the numerical models. The resultsof this study also provide useful information for the suitableapplications of the two models in practical surface scatteringcalculations.II. CONFIGURATIONFig. 1 shows the configuration of the study where rεis thecomplex dielectric constant of the surface and iθ is theincident angle. A 2-D (3-D scattering problem) isotropicGaussian correlated surface with surface rms height σandsurface correlation length l is considered.Fig 1. Surface configurationIII. SURFACE SCATTERING MODELSa. Small Slope Approximation (SSA) ModelThe Small Slope Approximation (SSA) [3] is chosen forthe comparison as recent studies have shown that this is arobust model for surface scattering. The SSA is based on aseries expansion in terms of surface “quasi-slope”, withevaluation of average incoherent cross sections from the firstterm (first order in slope) requiring a calculation similar tothat of the Kirchhoff approximation but retaining agreementwith perturbation theory in the small height limit.Expressions for average cross sections accurate to secondorder in slope are also available [3] but require additionalintegrations. An alternate approach for evaluation of higherorder corrections is described in [4] based on Monte Carlosimulation. The Monte Carlo procedure is applied in thispaper, with averaged second order in slope cross sectionsgenerated from 100 realisations of 128 by 32 lambda periodiciθrεl,σsurfaces sampled into 1024 by 256 points under plane waveincidence. Because the higher order SSA calculation can beimplemented with the Fourier transform, the computation isvery efficient compared to a Monte Carlo simulation with theMethod of Moments.b. Integral Equation Model (IEM)For the Integral Equation Model (IEM) in [2], it is knownthat the model provides a good prediction of surfacescattering coefficients for a wide range of surface profileswhich include the limits of both the classical Kirchhoffmodel (KM) and the small perturbation model (SPM). In theexpression of the IEM model, three terms of surfacescattering contributions, namely the Kirchhoff term (k), thecross term (kc) and the complementary term (c) are includedand shown in (1), respectively:cqpkcqpkqpoqpσσσσ++= (1)where 0qpσ is the surface backscattering coefficient and thesubscripts q and p denote the scattered polarisation and theincident polarisation, respectively. In this study, animproved model of IEM is used where this model is includedwith the empirical reflection coefficient model [5] thatprovides smooth transition in the Fresnel reflectioncoefficient approximation across a range of angular andfrequency conditions.c. Method of Moments (MOM)Approximate model results are compared with an “exact”numerical model based on Monte Carlo simulation with aniterative method of moments based surface scatteringcomputation. Computational time for the numerical model isreduced through the use of the “canonical grid” algorithm forcomputing matrix-vector multiplies to order (N log N) whereN is the number of surface sampling points. A detaileddescription of the algorithm is provided in [6]-[7]. To furtherkeep computational times reasonable, numerical results use32 realisations of 16 x 16 lambda surfaces sampled into 128by 128 points, and the Monte Carlo simulation is performedthrough the use of a IBM P2SC parallel computing resourcesat the Maui High Performance Computing Center [8]. Due tothe finite size of the surfaces modeled in the MOM, a“tapered-wave” incident field is used to avoid artificialsurface edge scattering effects. The tapered wave and surfacesize chosen make MOM results accurate only to up toincidence angle 50 degrees, so the comparison is notcontinued outside this range. Numerical model results arealso not included for incidence angle 0 degrees due to thedominance of the coherent scattered field at this angle for thesurface statistics considered.IV. RESULTS AND DISCUSSIONIn this study, a Gaussian correlated surface with complexdielectric constant 4+j1 is used. A surface profile with σk =0.5, kl = 3.0 is chosen where k is the wave number.