Models for Scattering from Rough Surfaces (Electromagnetic Waves) Part 2

The Kirchhoff approximation

In this section we shall consider the Kirchhoff (also sometimes referred to as the tangent plane approximation) approach to describe the scattering from rough surfaces, which was one of the first methods applied. We will consider surfaces with random surface profiles (i.e. not period surfaces) and within the context of the vector theory we will discuss the Kirchhoff Approximation. We will consider here the case of scattering from 2-dimensional dielectric surfaces. We will present results for the case of a surface which can be characterised as a Gaussian random process. We will also mention some extensions of the Kirchhoff approximation and will give references to further reading about the Kirchhoff approach. The reference list is by no means complete, since the literature on the Kirchhoff approximation is vast. A good representation of the Kirchhoff method can be found for instance in (Tsang et al, 2000, Tsang & Kong, 2001, Ulaby et al, 1982).

Formulation of the scattering problem

The geometry of the scattering problem we consider is shown in figure 2.4.1. We consider a monochromatic, linearly polarised incident plane wave with electric and magnetic field given by the equations (2.1.14) and (2.1.15), respectively.

It can be shown, similarly to equation (2.2.10), that the far zone scattering field, E^, can be written in terms of the tangential surface fields in the medium above the separating surface as (Stratton-Chu integral) (Ulaby et al, 1982):

where

What needs to be calculated are the tangential surface fields in equation (3.1.1). In equations (2.2.11) – (2.2.12) and (2.2.15) – (2.2.16) we presented integral equations for the tangential surface fields in the medium above the scattering dielectric surface. It should be noted that these expressions are exact. However, they cannot in general be solved analytically and therefore approximations have to be introduced. Below we will show that by introducing an approximation called the tangent plane approximation (or the Kirchhoff approximation), closed analytical solutions can be obtained to the scattering problem.

The tangent plane approximation and the Kirchhoff fields

In the Kirchhoff approach, the total fields at any point of the surface (i.e., the incident plus the scattered one, to be considered inside the integral (3.1.1)) are approximated by the fields that would be present on an infinitely extended tangent plane at that particular point on the surface. The reflection is therefore considered to be locally specular. It is due to this fact that the Kirchhoff approximation is also referred to as the tangent plane approximation. The Kirchhoff approach requires to be valid that every point on the surface has a large radius of curvature relative to the wavelength of the incident field.

Thus, under the tangent-plane approximation, the total field at a point on the surface is assumed equal to the incident field plus the field reflected by an infinite plane tangent to the point. Hence, the tangential surface fields are (Ulaby et al, 1982):

Here the subscript k stands for the Kirchhoff approximation.

The way to proceed from here, in most presentations of the Kirchhoff method, consists in expressing the tangential fields under the Kirchhoff approximation in terms of the incident electric field components and the local Fresnel reflection coefficients, which depend on the local angles of incidence. This results in the following expressions:

where the unit vectors t, d, ki define the local reference coordinate system (see (Fung, 1994)) and n is the unit normal vector to the interface in the above medium. Rv and Rh are the Fresnel reflection coefficients for vertical and horizontal polarisation respectively. Upon substituting (3.2.3) and (3.2.4) in (3.1.1), the scattered field is:

of the incident wave has been pointed out from the where the phase factor, of the incident wave has been pointed out from the equations (3.2.3) and (3.2.4). Such equation represents the scattered field formulated under the tangent-plane, or Kirchhoff approximation. As it stands the expression is a complicated function of the surface function and its partial derivatives. No analytic solution has been obtained from (3.2.5) without additional simplifying assumptions. Here we will show the results presented in (Ulaby et al, 1982): for surface with large (with respect to wavelength) standard deviation of surface heights, for which the stationary-phase approximation (Geometric Optics, GO) will be used, and for surfaces with small slopes and a medium or small standard deviation of surface heights, for which a scalar approximation (Physical Optics, PO) will be used.

The scattered field under the stationary-phase approximation (Geometric Optic, GO)

Under the stationary-phase approximation the local tangent plane on a surface point can be considered infinitely wide and, as consequence, the angular re-irradiation pattern originating from that specific point can be represented by a delta function centred in the specular direction. This means that scattering can occur only along directions for which there are specular points on the surface. Hence local diffraction effects are excluded. The approximating relations are obtained from the phase Q of (3.2.5), that is:

where

The phase Q is said to be stationary at a point if its rate of change is zero at the point, that is:

Hence, the partial derivatives of the surface slopes can be replaced by the components of the phase as:

Since, the local unit vector n is a function of the surface derivatives:

the use of (3.2.1.7) and (3.2.1.8) makes n x E and n x H independent on the integration variables. Thus, the expression for Es can be rewritten as:

where

The scattering field corresponding to transmission of p polarisation and reception of q polarisation can be written as (Ulaby et al, 1982):

where

To compute the scattering coefficient, defined in (2.4.1), for different polarisation states, it is necessary to calculate the ensemble average of I1 :

By assuming the surface roughness as a stationary and isotropic Gaussian random process, with zero mean, variance a2 , and correlation coefficient p, and in the assumption that the standard deviation of surface heights is large (that is, (qzc) large) the integral can be solved. The result is (Ulaby et al, 1982):

where the illuminated area A0 is (2L )2, p"(0) is the second derivatives of pevaluated at the origin and c2| p"(0)| corresponds to the mean-squared slope of the surface (Ulaby et al, 1982) (Section 2.3.1).

Upon substituting (3.2.1.15) into the product in the scattered-field expression, it follows:

Substituting (3.2.1.16) in the definition of the scattering coefficient given by equation (2.4.1), it assumes the following expression:

In the derivation of a p , the effects of shadowing and multiple scattering have been ignored. It is important to underline that (3.2.1.17) is valid only for surface with sufficiently large standard deviation of surface heights. Under such assumption, that is (qza)2 large, the scattering is purely incoherent. As (qzc) decreases, some scattered energy begins to appear in the coherent component. To examine such situation, a different approximation to the tangential fields is needed to permit small (qza )2 . This is discussed in the next section.

The scattered field under the scalar approximation (Physical Optics, PO)

A different Kirchhoff approach is the Physical Optics solution to (3.1.1). The Physical Optics approach involves the integration of the Kirchhoff scattered field over the entire rough surface, not just the portions of surface which contribute specularly to the scattered direction. Unlike the Geometric Optics solution, the Physical Optics solution predicts a coherent component.

The power in the incoherent reflected field can be found by expanding the Stratton-Chu equation in a Taylor series in surface slope distribution. In (Ulaby et al, 1982) the Physical Optics solution is called scalar approximation because slopes are ignored in the surface coordinate system, leading to a decoupling of polarisation in the vector scattering equations. Accordingly, the basic scattered-field expression can be rewritten in the form:

for the scattering-coefficient where Uqp are given in (Ulaby et al, 1982). To find for the scattering-coefficient computation, the following integral needs to be computed:

Since all U qp are expressed in a Taylor series in surface slope distribution, Zx and Zy:

can be written up to where ai are polarisation-dependent coefficients, the product can be written up to the first order in slope as:

Since (qzc)2 is no longer required to be large and assuming the size of the illuminated area equal to 2L x 2L , the ensemble average of the first term in (3.2.2.4) can be expressed as (for more details see (Ulaby et al, 1982))

where the n = 0 term corresponds to coherent scattering. It can be shown that this coherent-scattering coefficient can be expressed as:

which shows that coherent scattering is important only when qzc is small. The rest of the series in (3.2.2.5) represents incoherent scattering. The integral I0 for n > 1 can be rewritten in the following manner pointing out the illuminated area

For an isotropically rough surface with correlation length l and Gaussian normalised autocorrelation function,

the integral (3.2.2.7) can be shown to be:

It is clear that different solutions may be obtained for the integral if the normalised surface autocorrelation function is assumed to take some other functional forms. Upon substituting (3.2.2.7) and (3.2.2.8) into the factor (e^E^*} , the scattering coefficient for the incoherent part of the a0 term has the following expression:

 

If the normalised surface autocorrelation is not known, can be written as:

An additional contribution to the total scattering coefficient comes from the slope terms in (3.2.2.4). It can be computed taking into account in the ensemble average (ESqVESqV*^j the integrals of the slope terms in the x- and y-direction. The results of such integrals for a Gaussian normalised autocorrelation function are reported in (Ulaby et al, 1982). Also the expressions of the polarisation-dependent coefficients ai can be found in the same reference. However, the expressions of the coefficient a0 for each polarisation are reported below for the two particular cases of backscattering and scattering in the specular direction.

In the backscattering:
HH polarisation:
VH polarisation:
VV polarisation:
HV polarisation:
Conversely, in the specular direction case:
HH polarisation:
VH polarisation:
VV polarisation:
HV polarisation:

The quantity qx,y,z are defined in the previous section.

On the range of validity of the Kirchhoff method and shadowing effects

The basic assumption of the Kirchhoff method is that plane-boundary reflection occurs at every point on the surface. Thus, when statistical surfaces are considered, their horizontal-scale roughness, the correlation length l, must be larger than the electromagnetic wavelength, while their vertical-scale roughness, the standard deviation a of surface heights, must be small enough so that the average radius of curvature is larger than the electromagnetic wavelength. Mathematically, for stationary isotropic Gaussian surface the above-stated restriction are (Ulaby et al, 1982):

where k is the wave number and X is the electromagnetic wavelength. Note that the surface standard deviation should be small relative to the correlation length, but it can be comparable to or even larger than the electromagnetic wavelength. This means that large standard deviations can be tolerated if the correlation length is large enough to preserve an acceptable average radius of curvature. The conditions reported above are for the Kirchhoff approximation. The scattering models described in section 3.2.1 and 3.2.2 require additional approximations reported in the following table:

Validity limits of Kirchhoff (Gaussian sur	Approximation (KA) face)
Stationary Phase Aproximation (GO)	Scalar Approximation (PO)

Table 3.3.1. Validity of GO and PO for stationary isotropic Gaussian surfaces with standard deviation a and correlation length l.

Some concluding remarks on the Kirchhoff method

As was mentioned in the previous paragraph, the Kirchhoff method does neither in itself account for shadowing and nor does it (in the form described here) account for multiple scattering on the surface. Due to the lack of these two effects energy conservation is not satisfied. However, in (Ulaby et al, 1982) this conservation is demonstrating with the inclusion of these two effects.

In the literature, the surface height distribution is in most cases assumed to be Gaussian. The reason for this is, as mentioned previously, that the surface roughness rms height and the autocorrelation function entirely determine the random process, and therefore the bistatic scattering coefficient can be expressed in terms of these two quantities.

The Kirchhoff method has been applied to surfaces described by fractal geometry. As an example we can mention that in (Franceschetti et al, 1999) a fractional Brownian motion model was used for modelling the scattering from natural rough surface. In combination with the Kirchhoff method an analytical solution for the bistatic scattering coefficient was obtained.