Introduction
Models for scattering of electromagnetic waves from random rough surfaces have been developed during the last two centuries and the scientific interest in the problem remains strong also today due to the importance of this phenomenon in diverse areas of science, such as measurements in optics, geophysics, communications and remote sensing of the Earth. Such models can be categorised into empirical models, analytical models and a combination of the two. Though very simple, empirical models are greatly dependent on the experimental conditions. In spite of their complexity, only theoretical models can yield a significant understanding of the interaction between the electromagnetic waves and the Earth’s surface, although an exact solution of equations governing this interaction may not always be available and approximate methods have to be used. The semi-empirical models, which are based on both physical considerations and experimental observations, can be set between these two kinds of models and can be easily inverted. In this survey, we will focus on the analytical models and we study more in detail the Kirchhoff Approximation (KA), the Small Perturbation Method (SPM) and the Integral Equation Method (IEM). The Kirchhoff Approximation and the Small Perturbation Methods represent early approaches to scattering which are still much used, whereas the Integral Equation Method represents a newer approach which has a larger domain of validity. These methods have been found to be the most common in the literature and many of the other methods are based or have much in common with these approaches. In section 2, we begin by giving a brief presentation of the scattering problem and introduce some concepts and results from the theory of electromagnetic fields which are often used in this context. We will also define the bistatic scattering coefficient, due to the importance of this type of measurement in many remote sensing applications, and in particular in the retrieval of soil moisture content. In section 3, we give a brief presentation on the Kirchhoff Approximation and its close variants, the Physical Optics (PO) and the Geometrical Optics (GO). In section 4, we give a brief presentation of the Small Perturbation Method and in section 5 we will present the Integral Equation Model.
Some concepts of the electromagnetic theory and surface parameters
In this section we will give a brief presentation of some concepts on theories of electromagnetism and statistical characterisation of surfaces, which are often used for modelling scattering of electromagnetic waves from random rough surfaces. We will also define the bistatic scattering coefficient due to the importance of this type of measurement in many remote sensing applications.
The Maxwell’s equations and the wave equation
The basic laws of the electromagnetism are given by the Maxwell’s equations which, for linear, homogeneous, isotropic, stationary and not dispersive media, can be written as (Balanis, 1989):
where E is the electric field vector, D is the electric flux density, H is the magnetic field vector, B is the magnetic flux density, J is the conduction electric current density, Ji is the impressed electric current density and p is the electric charge density. Maxwell’s equations together with the boundary conditions, give a complete description of the field vectors at any points (including discontinuities) and at any time. In rough surface scattering, the surface enters in the boundary conditions (see equations (2.2.1)-(2.2.4)), which have to be also supplied at infinity.
If we consider time-harmonic variation of the electromagnetic field, the instantaneous field vectors can be related to their complex forms. Thus the Maxwell’s equations can be written in a much simpler form:
where we assumed the region characterised by permeability j, permittivity s and conductivity c (lossy medium). To obtain the governing equation for the electric field, we take the curl of (2.1.5) and then replace (2.1.6). Thus,
which is known as the inhomogeneous Helmholtz vector wave equation. In a free-source region, V- E = 0 and (2.1.9) simplifies to:
In rectangular coordinates, a simple solution to (2.1.10) has the form:
where E0 is a constant complex vector which determines the polarisation characteristics and the complex propagation vector, k , is defined as:
with the components satisfying
Equation (2.1.11) represents a plane wave and k is the propagation constant. Most analytical methods for scattering from rough surfaces assume this kind of incident wave, which if linearly polarised can be rewritten as:
where ki = kik, p is the unit polarisation vector and E0 is the amplitude. The associated magnetic field is given by:
where
is the wave impendence in the medium.
Integral theorems and other results used in scattering models
We will present some results for electromagnetic fields which are often used as a starting point in the analytical models for scattering from rough surfaces. These equations are approximated and simplified using different methods and assumptions in the analytical solutions for scattering from rough surfaces. We will not show how the equations in this section are derived, but derivation can be found in the references.
Consider an electromagnetic plane wave incident on a rough surface as shown in figure 2.2.1.
Fig. 2.2.1. Scattering of electromagnetic field on surface separating two media.
Across any surface interface, the electromagnetic field should satisfy continuity conditions given by (Balanis, 1989):
where n is the unit normal vector of the rough surface (pointing in the region 0). The electric surface current density, Js, and the charge surface density, p, at the rough interface are zero unless the scattering surface (or one of the media) is a perfect conductor. Using the fact that the fields satisfy the Helmholtz wave equation (2.1.9), it can be shown that in the region 0, the electromagnetic fields E and H, satisfy Huygens’ principle and the radiation boundary condition at infinity and E is given by (Ulaby et al, 1982; Tsang et al, 2000):
where G is the dyadic Green function (to the vector Helmholtz equation) which is represented by:
Here I is the unit dyadic and g (r, r’) is the Green function that satisfies the scalar wave equation. It assumes the following expression:
In (2.2.5) the first term on the right-hand side represents the field generated by a current source in an unbounded medium with permittivity s and permeability j and corresponds to the incident field. Hence, the electromagnetic field in the region 0 is expressed as the sum of two contributions: one is given by the incident field Ei (r); the other contribution is given by the surface integrals that involve the tangential components Et and Ht of the fields at the boundary S1 (note that n’ x E = n’ x Et and n’ x H = n’ x Ht) and represents the scattered field due to the presence of surface.
The equation (2.2.5) constitutes the mathematical basis of Huygens’ principle in vector form. According to this principle, the electromagnetic field in a source-free region (J = 0) is uniquely determined once its tangential components are assigned on the boundary of the region. However, since in the region 0, the existence of the impressed current J has been assumed, the total electric field can be expressed as the sum of two terms, the incident and scattering ones:
Thus, the scattered field can be written as:
If the observation point is in the far field region, the Green function in (2.2.9) can be simplified and the scattering field can be written as (Ulaby et al,1982; Tsang et al, 2000):
is the unit vector pointing in the direction of observation.
is the unit vector pointing in the direction of observation.
The tangential surface fields n x E and n x H can be also expressed as (Poggio & Miller, 1973):
and
where
and n , n’, nt, n’t are the unit normal vectors to the surface and nt = -n, n’t = -ri’, ri x E and n x H are the total tangential fields on the rough surface in the medium above the separating interface; G1 and G2 are the Green’s functions in medium above and below the interface, respectively, and
The nature of surface scattering
When an electromagnetic wave impinges the surface boundary between two semi-infinitive media, the scattering process takes place only at the surface boundary if the two media can be assumed homogeneous. Under such supposition, the problem at issue is indicated as surface scattering problem. On the other hand, if the lower medium is inhomogeneous or is a mixture of materials of different dielectric properties, then a portion of the transmitted wave scattered backward by the inhomogeneities may cross the boundary surface into the upper medium. In this case scattering takes place within the volume of the lower medium and it is referred to as volume scattering. In most cases both the scattering processes are involved, although only one of them can be dominant. In the case of bare soil, which will be assumed to be a homogeneous body, surface scattering is the only process taken into consideration. When the surface boundary separating the two semi-infinitive media is perfectly smooth the reflection is in the specular direction and is described by the Fresnel reflection laws. On the other hand, when the surface boundary becomes rough, the incident wave is partly reflected in the specular direction and partly scattered in all directions. Qualitatively, the relationship between surface roughness and surface scattering can be illustrated through the example shown in Figure 2.3.1. For the specular surface, the angular radiation pattern of the reflected wave is a delta function centred about the specular direction as shown in Figure 2.3.1 (a). For the slightly rough surface (Figure 2.3.1 (b)), the angular radiation pattern consists of two components: a reflected component and a scattered component. The reflected component is again in the specular direction, but the magnitude of its power is smaller than that for smooth surface. This specular component is often referred to as the coherent scattering component. The scattered component, also known as the diffuse or incoherent component, consists of power scattered in all directions, but its magnitude is smaller than that of the coherent component. As the surface becomes rougher, the coherent component becomes negligible. Note that the specular component represents also the mean scattered field (in statistical sense), whereas the diffuse component has a stochastic behaviour, associated to the randomness of the surface roughness.
Fig. 2.3.1. Relative contributions of coherent and diffuse scattering components for different surface-roughness conditions: (a) specular, (b) slightly rough, (c) very rough.
Characterisation of soil roughness
A rough surface can be described by a height function g = z (x, y). There are basically two categories of methods which are being used to measure surface roughness. The roughness can be carried out by means of various experimental approaches able to reproduce the surface profile by using contact or laser probes, or it can be estimated using some theory which relates scattering measurements to surface roughness. In general, the study of scattering in remote sensing is performed by using random rough surface models, where the elevation of surface, with respect to some mean surface, is assumed to be an ergodic1, and hence stationary2, random process with a Gaussian height distribution.
Accordingly, the degree of roughness, or simply the roughness, of a random surface is characterised in terms of statistical parameters that are measured in units of wavelength. For this reason, a given surface that may "appear" very rough to an optical wave, may "appear" very smooth to a microwave.
The two fundamental parameters commonly used are the standard deviation of the surface height variation (or rms height) and the surface correlation length. Such parameters describe the statistical variation of the random component of surface height relative to a reference surface, that may be the unperturbed surface of a period pattern, as in the case of a row-tilled soil surface (Figure 2.3.1.1. (a)), or may be the mean plane surface if only random variations exist (Figure 2.3.1.1 (b)).
Fig. 2.3.1.1. Two configurations of height variations: (a) random height variations superimposed to a periodic surface; (b) random variations superimposed to a flat surface.
Let z (x) be a representative realisation of the ergodic and stationary process that describes a generic rough surface in a one-dimensional case. The mean value, which throughout this topic will be denoted by angular brackets (…) , is equal to the spatial average over a statistically representative segment of the surface, of dimensions Lx, centred at the origin:
As it can be noted from the above definition, for a stationary surface the average does not depend on x. The second moment is:
1 A process is ergodic when one realisation is representative of all the process, i.e. the statistical averages over an extracted random variable may be replaced by spatial averages over a single realisation.
2 The stationarity implies that all the statistically properties of a random process are invariant under the translation of spatial coordinates.
1 A process is ergodic when one realisation is representative of all the process, i.e. the statistical averages over an extracted random variable may be replaced by spatial averages over a single realisation.
2 The stationarity implies that all the statistically properties of a random process are invariant under the translation of spatial coordinates.
Using the above expressions, the standard deviation of the surface height, c, is therefore defined as:
Such quantities characterise the dispersion of the surface height relative to the reference plane. Taking into account the stationary properties of the process and considering its mean value null, the variance, c2 , is coincident with the second moment and does not depend on x. The autocorrelation function of the height random process z (x) is given by:
The normalised autocorrelation function (ACF), better known as the correlation coefficient, assumes for a process with zero mean value the following expression:
It is a measure of the similarity between the height z at point x and at point distant r from x. It has the following properties:
The spectral density or power spectrum is defined, for an ergodic random process, as the Fourier transform of the autocorrelation function Rz (x):
where kx is the Fourier transform variable.
However, taking into account the equation (2.3.1.5), it is common practice in characterising the random surface to define the power spectrum of the normalised autocorrelation function:
The Gaussian distribution plays a central role in modelling scattering from random rough surfaces because it is encountered under a great number of different conditions and because Gaussian variates have the unique property that the random process is entirely determined by the height probability distribution and autocorrelation. All higher order correlations can be expressed in terms of the (second order) autocorrelation function, which simplifies modelling the surface scattering process. A simple and often used form for the autocorrelation is the Gaussian function but other forms have also been studied (Saillard & Sentenac, 2001).
The roughness spectrum at the n’th power of the autocorrelation function, W(n), which often enters into closed form solutions of the scattering problem, is given by the Fourier transform:
The consideration of a realistic autocorrelation function is in fact a relevant problem for a better modelling of the soil scattering. Some often used forms (see for instance (Fung, 1994)) of the autocorrelation function are the Gaussian correlation function, the exponential correlation function, combinations of the Gaussian and exponential functions and the so called 1.5-power correlation function. For all of these, the roughness spectrum at the n’th power can be evaluated analytically (see (Fung, 1994)). For instance, for an isotropically rough surface, the normalised Gaussian autocorrelation in a single dimension assumes the following expression:
where l is the correlation length. Such surface parameter is defined as the displacement x for which p(x) is equal to 1/e
The correlation length of a surface provides a reference for estimating the statistical independence of two points on the surface; if the two points are separated by a horizontal distance greater than l, then their heights may be considered to be (approximately) statistically independent of one another. In the extreme case of a perfectly smooth (specular) surface, every point on the surface is correlated with every other point with a correlation coefficient of unity. Hence, l = <n in this case.
Referring to equation (2.3.1.9), the n’th power roughness spectrum is equal to:
Beside the height random function z (x), the slope function is another important characterisation of the rough surface. It is defined as:
Considering the stationary random process z (x) as normally distributed with zero mean and variance c2 , being Zx the first derivative, its distribution is again normal with zero mean and variance related to the second derivative of the autocorrelation function of z (x) at the origin (Beckman & Spizzichino, 1963):
The rms slope is subsequently indicated as m:
When the normalised autocorrelation function is Gaussian (equation (2.3.1.9)), the rms slope is equal to:
Bistatic scattering coefficient
A quantity often used in models and measurements of scattering in the microwave region is the bistatic scattering coefficient a° p6j,0S, js). It describes the target’s scattering properties at a given frequency, polarisation, incidence and observing directions, being independent on the specific measurement system used. It is possible to define a^ p directly in terms of the incident and scattering field Eip and Eqs as follows (Ulaby et al, 1982):
where the ensemble average must be considered in case the scattered field is the fluctuating zero mean component (i.e., the diffuse or incoherent component mentioned before) generated by a natural target or random rough surface. Such equation shows a^ p as the ratio of the total power scattered by an equivalent isotropic scatterer in direction (0S ) to the product of the incident power density in direction 6 ) and the illuminated area. The backscattering coefficient a°p(6) is a special case ofa^ p(6{j6j); it is defined for 6S = 6i and js = j{ ±n (Figure 2.4.1), which corresponds to the incident and scattered direction being the same except for a reversal in sense.
Fig. 2.4.1. Geometry of the scattering problem.
