Models for Scattering from Rough Surfaces (Electromagnetic Waves) Part 3

The small perturbation method

The Small Perturbation Method (SPM) belongs to a large family of perturbation expansion solutions to the wave equation. The approach is based on formulating the scattering as a partial differential equation boundary value problem. The basic idea is to find a solution in terms of plane waves that matches the surface boundary conditions, which state that the tangential component of the field must be continuous across the boundary. The surface fields are expanded in a perturbation series with respect to surface height, e.g., E = E0 + Ea +…. In the expansion E0 would be the surface field if the surface was flat. The philosophy behind this approach is that small effective surface currents on a mean surface replace the role of a small-scale roughness. So this method applies to surfaces with small surface height variations and small surface slopes compared with the wavelength but independently of the radius of curvature of the surface. Therefore, the surface needs no longer to be approximated by planes. The small-scale roughness is expanded in a Fourier series and the contribution to the field is therefore analysed in terms of different wavelength components.

Here we will report only the expressions of the bistatic scattering coefficient. A more detailed description of their computation process can be found in (Ulaby et al, 1982).

A small presentation of the SPM

The zero order solution of the SPM is the same as for a plane interface, while the first order solution gives the incoherent scattered field due to single scattering. For the latter case, the bistatic scattering coefficient for either a horizontally or vertically polarised incident wave is (Ulaby et al, 1982):

where

c and p(u, v) are, respectively, the variance of surface heights and the surface correlation coefficient; aqp are coefficients that depend on polarisation, incidence and scattering angle, and on complex relative dielectric constant sc of the homogeneous medium below the interface. The detailed expressions of aqp are reported in (Ulaby et al, 1982).

Some remarks on the region of validity of the SPM

The Small Perturbation Method is applied to surfaces with a surface height standard deviation much less than the incident wavelength (5 percent or less) and an average surface slope comparable to or less than the surface standard deviation times the wave number. For a surface with Gaussian correlation function, such two conditions can be expressed analytically as follows, but they should be viewed only as a guideline for applying the SPM scattering model:

The SPM has been compared to more accurate numerical simulations in (Thorsos & Jackson, 1989; 1991) for one-dimensional rough surfaces with a Gaussian roughness spectrum. Under these conditions the authors show that the first-order SPM gives accurate results for ka << 1 and kl«1. The results also show that for ka << 1 and kl > 6, the sum of the first three orders of the SPM is required to obtain accurate results.

It has been argued that the SPM does account for multiple scattering up to the order of the perturbative expansion. This means that the first order perturbative solution does not account for multiple scattering but that some multiple scattering effects can be observed in the higher order solutions.

Validity limits of Smal

(Gaus

l Perturbation Method (SPM)

sian surface)

Table 4.2.1. Validity of SPM for stationary isotropic Gaussian surfaces with standard deviation a and root mean square slope rmsslope.

The Integral Equation Method (IEM)

A relatively new method for calculating scattering of electromagnetic waves from rough surfaces is the Integral Equation Method (IEM). The IEM has been used extensively in the microwave region in recent years and it has proved to provide good predictions for a wide range of surface profiles. The method can be viewed as an extension of the Kirchhoff method and the Small Perturbation Method since it has been shown to reproduce results of these two methods in appropriate limits. The IEM is a relatively complicated method in its general form (including multiple scattering) and it is beyond the scope of the present overview to give a full presentation of the method. A more detailed presentation of the IEM can be found in (Fung, 1994).

On the formulation of the IEM

The starting point of the IEM is the Stratton-Chu integral for the scattered field, equation (3.1.1). The tangential surface fields which enter the Stratton-Chu integral are given in equations (2.2.11) – (2.2.12) and (2.2.15) – (2.2.16). In the Kirchhoff approach, the tangential fields are approximated using the tangent plane approximation, replacing the complete tangential surface fields with the Kirchhoff tangential surface fields of equations (3.2.1) and (3.2.2). It is clear that the Kirchhoff tangential surface fields cannot provide alone a good estimate of the surface fields since the integral form in equations (2.2.11) – (2.2.12) are not accounted for in the Kirchhoff approach. In the IEM, a complementary term is included in equations (3.2.1) and (3.2.2) to correct for this:

In these equations, the first terms on the right hand side are the tangential fields under Kirchhoff approximation and the complementary fields are given by:

Er and Hr being the reflected electric and magnetic fields propagating along the reflected direction. To use (5.1.1) and (5.1.2) for estimating the tangential field, both the Kirchhoff field and the complementary field need to be expressed in terms of the incident field components and the surface reflectivity properties. Using the local coordinate system defined by the unit vectors t, d, Iti (for their expressions refer to (Fung, 1994)), the incident electric and magnetic field can be expressed into locally horizontally and vertically polarised components. Accordingly, after some manipulations (see (Fung, 1994) for more details), the Kirchhoff and complementary tangential fields can be rewritten as:

It can be noted that, while (5.1.5) and (5.1.6) are expressed in terms of known quantities, that is the incident electric or magnetic fields, the local Fresnel reflection coefficient and the local incident angle, (5.1.7) and (5.1.8) are integral equations. In order to obtain estimates of (5.1.7) and (5.1.8), IEM substitutes the unknown expressions of the tangential fields in the right-hand side of (5.1.7) and (5.1.8), that is the (n’x E’) and (n’x H’) terms which appear in e, et, H and Ht, with the Kirchhoff tangential fields, (n’x E’) and (n’x H’) , respectively.

This is the fundamental approximation adopted by IEM model. However, even with this simplification the obtained integral expressions remain too complex for practical use.

Much simpler approximate expressions of the tangential Kirchhoff and complementary fields can be obtained differentiating them for each linear incident and scattered polarisation. The resulting approximated equations (electric and magnetic surface field equations for horizontal, vertical and cross polarisation) can be found in (Fung, 1994). Then, the simplified tangential surface fields can be inserted in the Stratton-Chu integral. The far field scattered from the rough surface can be expressed as a combination of the Kirchhoff and the complementary term:

where

and

The quantities fqp and Fqv, respectively the Kirchhoff and complementary field coefficients, that appear in the above equations are defined as follows:

and Ei is the complex amplitude of the incident electric field.

In general, both fqp and Fqv are dimensionless, complicated expressions and depended on spatial variables. Therefore several approximations are made to make these functions independent of spatial variables (Fung, 1994).

In particular, the fqp coefficients depend on the Fresnel reflection coefficients, and hence on the local angle, and on the slope terms, Zx and Zy. The first dependency is removed by approximating the local incidence angle in the Fresnel reflection coefficients by the incident angle, 6, for surface with small scale roughness and by the specular angle, for surface with large scale roughness.The rule that defines the bound between the two regions is reported here assuming a Gaussian autocorrelation function:

In order to remove the dependence on the slope terms, the integral (5.1.10) is solved by parts and the edge terms were discarded.

To obtain the expressions of the complementary coefficients F, the computation is rather lengthy and complicated. When the equations (2.2.15) – (2.2.18) are substituted in the approximated expressions of tangential complementary fields, the spectral representations of Green’s function and of its gradient are introduced, assuming however the same Green’s functions for both the medium:

are the propagation vector and its z-component of the

are the propagation vector and its z-component of the generic plane wave that appears in the plane waves expansion of the field, whereas z and z’ are the random variables representing the surface heights at different locations on the random surface. In (Fung, 1994), the | z – z’| terms and the term with the + are dropped in the equations (5.1.15) and (5.1.16) in order to simplify the calculation. However, in an improved version of the IEM (see (Chen et al, 2000)) these terms are kept in the analysis. In addition, as was the case for the Kirchhoff coefficients, fqp, the dependence through the slope terms is removed by integrating by parts and discarding the edge terms. Instead, as regard the Fresnel reflection coefficients, the local angle is always replaced by incident angle (Fung, 1994; Wu et al, 2001).

Moreover, it is important to underline that the tangential and normal field components that appear in the expressions of the Fqp coefficients through equations (2.2.15) – (2.2.18) can be approximated by the tangential Kirchhoff fields. The complimentary field coefficients Fqp that appear in the right term of the equation (5.1.11) are obtained from the definition of the Fqp after the Green’s function and its gradient are replaced by the simplified spectral representation, above mentioned, and after the phase factor of the Green function and u, v, x’, y’ integrations are factored out. The expressions of such coefficients together with the expressions of the Kirchhoff ones are reported in (Brogioni et al, 2010).

Once the field coefficients, fqp and Fqp, are made independent of spatial variables, it is possible to provide the expression of the incoherent scattered power:

and from this the bistatic scattering coefficient:

From the above expression it follows that the scattering coefficient is given by the sum of three terms: the Kirchhoff, the complementary and the cross term. The first is originated by Kirchhoff fields, the second by the interaction between Kirchhoff and complementary fields, whereas the last is due only to complementary fields.

To carry out the average operation an assumption about the type of surface height distribution is necessary. In order to simplify the calculation of the incoherent power terms the rough surface is assumed characterised by a Gaussian height distribution. Accordingly, the terms in (5.1.18) assume the following expressions, reported in (Fung, 1994):

The above expressions consist of multiple integrals which are too complex and hence not practical to use. In order to evaluate these integrals, the model is approximated in two different forms depending upon whether the surface height is moderate or large in terms of the incident wavelength (ka). The first case is referred to as low frequency approximation, whilst the other is referred to as high frequency approximation. An indicative threshold value of ka < 2 is reported in (Fung, 1994). The detailed expressions ofvalid separately when ka< 2 and for large ka are given in (Fung, 1994) and are not reported here. For both the approximations, in the expression of the bistatic scattering coefficient two types of terms can be distinguished: one representing single-scattering and the other representing multiple-scattering. The latter may be viewed as a correction to the single term for both the high- and the low-frequency regions. This division is important to identify weather single or multiple scattering is significant for applications. For completeness we report here the total single scattering coefficient obtained by selecting the single scattering contributions in the expressions ofvalid when ka < 2 (for the detailed explanation refer to (Fung, 1994)):

Conclusions

We have presented the results from a literature search of models for scattering of electromagnetic waves from random rough surfaces. In particular we have focused on the calculation of the bistatic scattering coefficient in three different classes of methods: the Kirchhoff Approximation, the Method of Small Perturbation and the Integral Equation Method. Of these, the first two, are amongst the early approaches which however are still much used. The latter is an example of more recent approaches which have been developed as an attempt to extend the validity of the former methods.