1. Introduction

Each region of the Earth’s crust can be morphologically modeled as a suitable layered structure, in which some amount of roughness is presented by every interface. Actually, propagation in stratified soil, sand cover of arid regions, forest canopies, urban buildings, snow blanket, snow cover ice, sea ice and glaciers, oil flood on sea surface, and other natural scenes can be modeled referring to most likely discrete (piecewise-constant) systems, rather than continuous, with some amount of roughness presented by every interface. Moreover, a key issue in remote sensing of other Planets is to reveal the content under the surface illuminated by the sensors: also in this case a layered model is usually employed.

The aim of this chapter is to provide a structured presentation of the main theoretical and conceptual foundations for the problem of the electromagnetic wave interaction with layered rough media. In the first part, special emphasis is on the analytical models obtainable in powerful framework of the perturbation approach. The comprehensive scattering model based on the Boundary Perturbation Theory (BPT), which permits to systematically analyze the bi-static scattering patterns of 3D multilayered rough media, is then presented highlighting the formal connections with all the previously existing simplified perturbative models, as well as its wide relevance in the remote sensing applications scenario. The polarimetric Scattering Matrix of a multilayered medium with an arbitrary number of rough interfaces is also provided. The second part is devoted to a mathematical description which connects the concepts of local scattering and global scattering. Consequently, a functional decomposition of the BPT global scattering solution in terms of basic single-scattering local processes is rigorously established. The scattering decomposition gives insight into the BPT analytical results, so enabling a relevant physical-revealing interpretation involving ray-series representation. Accordingly, in first-order limit, the way in which the character of the local scattering processes emerges is dictated by the nature of the structural filter action, which is inherently governed by the series of coherent interactions with the medium boundaries. As a result, the phenomenologically successful BPT model opens the way toward new techniques for solving the inverse problem, for designing SAR processing algorithms, and for modelling the time-domain response of layered structures.

2. Problem definition

When stratified media with rough interfaces are concerned, the possible approaches to cope with the EM scattering problem fall within three main categories. First, the numerical approaches do not permit to attain a comprehensive understanding of the general functional dependence of the scattering response on the structure parameters, as well as do not allow capturing the physics of the involved scattering mechanisms. Layered structures with rough interfaces have been also treated resorting to radiative transfer theory (RT). However, coherent effects are not accounted for in RT theory and could not be contemplated without employing full wave analysis, which preserves phase information. Another approach relies on the full-wave methods. Although, to deal with the electromagnetic propagation and scattering in complex random layered media, several analytical formulation involving some idealized cases and suitable approximations have been conducted in last decades, the relevant solutions usually turn out to be too complicated to be generally useful in the remote sensing scenario, even if simplified geometries are accounted for. The proliferation of the proposed methods for the simulation of wave propagation and scattering in a natural stratified medium and the continuous interest in this topic are indicative of the need of appropriate modelling and interpretation of the complex physical phenomena that take place in realistic environmental structures. Indeed, the availability of accurate, sound physical and manageable models turns out still to be a strong necessity, in perspective to apply them in retrieving of add-valued information from the data acquired by microwave sensors. For instance, such models are high desirable for dealing with the inversion problem as well as for the effective design of processing algorithms and simulation of Synthetic Aperture Radar signals. Generally speaking, an exact analytical solution of Maxwell equations can be found only for a few idealized problems. Subsequently, appropriate approximation methods are needed. Regarding the perturbative approaches, noticeable progress has been attained in the investigation on the extension of the classical SPM (small perturbation method) solution for the scattering from rough surface to specific layered configurations. Most of previous existing works analyze different layered configurations in the first-order limit, using procedures, formalisms and final solutions that can appear of difficult comparison (Yarovoy et al., 2000), (Azadegan and Sarabandi, 2003), (Fuks, 2001). All these formulations, which refer to the case of a single rough interface, have been recently unified in (Franceschetti et al, 2008). On the other hand, solution for the case of two rough boundaries has also been proposed in (Tabatabaeenejad and Moghaddam, 2006).

Methodologically, we underline that all the previously mentioned existing perturbative approaches, followed by different authors in analyzing scattering from simplified geometry, imply an inherent analytical complexity, which precludes the treatment to structures with more than one (Fuks, 2001) (Azadegan et al., 2003) (Yarovoy et al., 2000) or two (Tabatabaeenejad er al., 2006) rough interfaces.

The general problem we intend to deal with here refers to the analytical evaluation of the electromagnetic scattering by layered structure with an arbitrary number of rough interfaces (see Figure 1). As schematically shown in Figure 1, an arbitrary polarized monochromatic plane wave

is considered to be incident on the layered medium at an angle θ0i relative to the z^ direction from the upper half-space, where in the field expression a time factor exp(-jt) is understood, and where, using a spherical frame representation, the incident vector wave direction is individuated by θ0i,ϕ0i

where k⊥i=kxix^+kyiy^ is the two-dimensional projection of incident wave-number vector on the plane z=0. The parameters pertaining to layer m with boundaries –d _m-1 and -d _m are distinguished by a subscript m. Each layer is assumed to be homogeneous and characterized by arbitrary and deterministic parameters: the dielectric relative permittivity _m , the magnetic relative permeability μ _m and the thickness _m =d _m -d _m-1 . With reference to Figure 1, it has been assumed that in particular, d ₀ =0. In the following, the symbol denotes the projection of the corresponding vector on the plan z=0. Here r=(r⊥,z) , so we distinguish the transverse spatial coordinates r⊥=(x,y) and the longitudinal coordinate z. In addition, each mth rough interface is assumed to be characterized by a zero-mean two-dimensional stochastic process ζm=ζm(r⊥) with normal vector n^m . No constraints are imposed on the degree to which the rough interfaces are correlated.

A general methodology has been developed by Imperatore et al. to analytically treat EM bistatic scattering from this class of layered structures that can be described by small changes with respect to an idealized (unperturbed) structure, whose associated problem is exactly solvable. A thorough analysis of the results of this theoretical investigation (BPT), which is based on perturbation of the boundary condition, will be presented in the following, methodologically emphasizing the development of the several inherent aspects.

3. Preliminary notation and definitions

This section is devoted preliminary to introduce the formalism used in the following of this chapter. The Flat Boundaries layered medium (unperturbed structure) is defined as a stack of parallel slabs (Figure 2), sandwiched in between two half-spaces, whose structure is shift invariant in the direction of x and y (infinite lateral extent in x-y directions). With the notations Tm−1|mp and Rm−1|mp , respectively, we indicate the ordinary transmission and reflection coefficients at the interface between the regions (m-1) and m, with the superscript p{v, h} indicating the polarization state for the incident wave and may stand for horizontal (h) or vertical (v) polarization (Tsang et al., 1985) (Imperatore et al. 2009a). In addition, we stress that:

The generalized reflection coefficients ℜm−1|mp , for the p-polarization (TE or TM), at the interface between the regions (m-1) and m are defined as the ratio of the amplitudes of upward- and downward-propagating waves immediately above the interface, respectively. They can be expressed by recursive relations as in (Chew W. C., 1997) (Imperatore et al. 2009a):

Likewise, at the interface between the regions (m+1) and m, ℜm+1|mp is given by:

where

where km=k0μmεm is the wave number for the electromagnetic medium in the mth layer, with k0=ω/c=2π/λ , and where k⊥=kxx^+kyy^ is the two-dimensional projection of vector wave-number on the plane z=0. It should be noted that the factors

take into account the multiple reflections in the mth layer. On the other hand, the generalized transmission coefficients in downward direction can be defined as:

where p {v, h}. The generalized transmission coefficients in upward direction are then given by:

which formally express the reciprocity of the generalized transmission coefficients for an arbitrary flat-boundaries layered structure (Imperatore et al. 2009b). In addition, with reference to a layered slab sandwiched between two half-space, we consider the generalized transmission coefficients in upward direction for the layered slab between two half-spaces (m,0), which are defined as

Note also that

The generalized transmission coefficients in downward direction for the layered slab between two half-spaces (m,0), can be defined as

On the other hand, it should be noted that the ℑ0|mp are distinct from the coefficients ℑ0|mp(slab) , because in the evaluation of ℑ0|mp the effect of all the layers under the layer m is taken into account, whereas ℑ0|mp(slab) are evaluated referring to a different configuration in which the intermediate layers 1...m are bounded by the half-spaces 0 and m. In the following, we shown how the employing the generalized reflection/transmission coefficient notions not only is crucial in obtaining a compact closed-form perturbation solution, but it also permit us to completely elucidate the obtained analytical expressions from a physical point of view, highlighting the role played by the equivalent reflecting interfaces and by the equivalent slabs, so providing the inherent connection between local and global scattering responses.

4. Spectral Representation of the Stochastic Geometry Description

In this section, the focus is on stochastic description for the geometry of the investigated structure, and the notion of wide-sense stationary process is detailed. First of all, when the description of a rough interface by means of deterministic function ζm(r⊥) is concerned, the corresponding ordinary 2-D Fourier Transform pair can be defined as

Let us assume now that ζm(r⊥) , which describes the generic (mth) rough interface, is a 2-D stochastic process satisfying the conditions

where the angular bracket denotes statistical ensemble averaging, and where Bζm(ρ) is the interface autocorrelation function, which quantifies the similarity of the spatial fluctuations with a displacement . Equations (17)-(18) constitute the basic assumptions defining a wide sense stationary (WSS) stochastic process: the statistical properties of the process under consideration are invariant to a spatial shift. Similarly, concerning two mutually correlated random rough interfaces ζm and ζn , we also assume that they are jointly WSS, i.e.

where Bζmζn(ρ) is the corresponding cross-correlation function of the two random processes.

It can be readily derived that

The integral in (15) is a Riemann integral representation for ζm(r⊥) , and it exists if ζm(r⊥) is piecewise continuous and absolutely integrable. On the other hand, when the spectral analysis of a stationary random process is concerned, the integral (15) does not in general exist in the framework of theory of the ordinary functions. Indeed, a WSS process describing an interface ζm(r⊥) of infinite lateral extension, for its proper nature, is not absolutely integrable, so the conditions for the existence of the Fourier Transform are not satisfied. In order to obtain a spectral representation for a WSS random process, this difficulty can be circumvented by resorting to the more general Fourier-Stieltjes integral (Ishimaru, 1978); otherwise one can define space-truncated functions. When a finite patch of the rough interface with area A is concerned, the space-truncated version of (15) can be introduced as

subsequently, ζ˜m(k⊥)=limA→∞ζ˜m(k⊥A) is not an ordinary function. Nevertheless, we will use again the (15)-(16), regarding them as symbolic formulas, which hold a rigorous mathematical meaning beyond the ordinary function theory (generalized Fourier Transform). We underline that by virtue of the condition (17) directly follows also that ζ˜m(k⊥)=0 . Let us consider

where the asterisk denotes the complex conjugated, and where the operations of average and integration have been interchanged. When jointly WSS processes ζm and ζn are concerned, accordingly to (19), the RHS of (22) must be a function of r′⊥−r″⊥ only; therefore, it is required that

where (∙) is the Dirac delta function, and where Wmn(κ) is called the (spatial) cross power spectral density of two interfaces ζm and ζn , for the spatial frequencies of the roughness. Equation (23) states that the different spectral components of the two considered interfaces must be uncorrelated. This is to say that the (generalized) Fourier transform of jointly WSS processes are jointly non stationary white noise with average power Wmn(k′⊥) . Indeed, by using (23) into (22), we obtain

where the RHS of (24) involves an (ordinary) 2D Fourier Transform. Note also that as a direct consequence of the fact that ζn(r⊥) is real we have the relation ζ˜n(k⊥)=ζ˜n*(−k⊥) . Therefore, setting ρ=r′⊥−r″⊥ in (24), we have

The cross-correlation function Bζmζn(ρ) of two interfaces ζm and ζn , is then given by the (inverse) 2D Fourier Transform of their (spatial) cross power spectral density, and Equation (25) together with its Fourier inverse

may be regarded as the (generalized) Wiener-Khinchin theorem. In particular, when n=m, (23) reduces to

where Wm(κ) is called the (spatial) power spectral density of nth corrugated interface ζm and can be expressed as the (ordinary) 2D Fourier transform of n-corrugated interface autocorrelation function, i.e., satisfying the transform pair:

which is the statement of the classical Wiener-Khinchin theorem. We emphasize the physical meaning of Wm(κ)dκ=Wm(κx,κ)ydκxdκy : it represents the power of the spectral components of the mth rough interface having spatial wave number between _x and _x +d _x and _y and _y +d _y , respectively, in x and y direction. Furthermore, from (20) and (26) it follows that

This is to say that, unlike the power spectral density, the cross power spectral density is, in general, neither real nor necessarily positive. Furthermore, it should be noted that the Dirac’s delta function can be defined by the integral representation

By using in (27) and (23) the relation δ(0;A)=A/(2π)2 , we have, respectively, that the (spatial) power spectral density of nth corrugated interface can be also expressed as

and the (spatial) cross power spectral density of two interfaces ζm and ζn is given by

It should be noted that the domain of a rough interface is physically limited by the illumination beamwidth. Note also that the different definitions of the Fourier transform are available and used in the literature: the sign of the complex exponential function are sometimes exchanged and a multiplicative constant (2π)−2 may appear in front of either integral or its square root in front of each expression (15)-(16). Finally, we recall that the theory of random process predicts only the averages over many realizations.

5. Perturbative Field Formulation

With reference to the geometry of Figure 1, in order to obtain a solution valid in each region of the structure, we have to enforce the continuity of the tangential fields:

where ΔEm=Em+1−Em ΔHm=Hm+1−Hm , and the surface normal vector is given by:

with the slope vector γm

and where ∇⊥ is the nabla operator in the x-y plane. In order to study the fields Em and Hm within the generic mth layer of the structure, we assume then that, for each mth rough interface, the deviations and slopes of surface with respect to the reference mean plane z=-d _m are small enough in the sense of (Ulaby et al, 1982) (Tsang et al., 1985), so that the fields can be expanded about the reference mean plane. The fields expansion can be then injected into the boundary conditions (34)-(35), so that, retaining only up to the first-order terms, the following nonuniform boundary conditions can be obtained (Imperatore et al. 2009a)

where the field solution has been formally represented as:

Therefore, the boundary conditions from each mth rough interface can be transferred to the associated equivalent flat interface. In addition, the right-hand sides of Eqs. (38) and (39) can be interpreted as effective magnetic ( JHmp(1) ) and electric ( JEmp(1) ) surface current densities, respectively, with p denoting the incident polarization; so that we can identify the first-order fluctuation fields as being excited by these effective surface current densities imposed on the unperturbed interfaces. Accordingly, the geometry randomness of each corrugated interfaces is then translated in random current sheets imposed on each reference mean plane (z=-d _m ), which radiate in an unperturbed (flat boundaries) layered medium. As a result, within the first-order approximation, the field can be than represented as the sum of an unperturbed part En(0),Hn(0) and a random part, so that En(r⊥z)≈En(0)+En(1) Hn(r⊥z)≈Hn(0)+Hn(1) . The first is the primary field, which exists in absence of surface boundaries roughness (flat-boundaries stratification), detailed in (Imperatore et al. 2009a); whereas En(1),Hn(1) can be interpreted as the superposition of single-scatter fields from each rough interface. In order to perform the evaluation of perturbative development, the scattered field is then represented as the sum of up- and down-going waves, and the first-order scattered field in each region of the layered structure can be represented in the form:

Therefore, a solution valid in each region of the structure can be obtained from (42)-(45) taking into account the non uniform boundary conditions (38)-(39). In order to solve the scattering problem in terms of the unknown expansion coefficients Sm±q(1)(k⊥) , we arrange their amplitudes in a single vector according to the notation:

Subsequently, the nonuniform boundary conditions (38)-(39) can be formulated by employing a suitable matrix notation, so that for the (q=h) horizontal polarized scattered wave we have (Imperatore et al. 2008a) (Imperatore et al. 2009a):

where

is the term associated with the effective source distribution, wherein the expressions of the effective currents J˜Emp(1) and J˜Hmp(1) , imposed on the (flat) unperturbed boundary z = −d _m , for an incident polarization p {v, h} are detailed in (Imperatore et al. 2009a); and where Z ₀ is the intrinsic impedance of the vacuum. Furthermore, the fundamental transfer matrix operator is given by:

with the superscripts q {v, h} denoting the polarization. Moreover, it should be noted that on a (kth) flat interface Eq. (47) reduces to the uniform boundary conditions, thus getting:

We emphasize that Eqs. (47) states in a simpler form the problem originally set by Eqs. (38)-(39): as matter of fact, solving Eq. (47) m implies dealing with the determination of unknown scalar amplitudes Sm±q(1)(k⊥) instead of working with the corresponding vector unknowns Em(1),Hm(1) . Therefore, the scattering problem in each mth layer is reduced to the algebraic calculation of the unknown expansion scattering coefficients vector (46). As a result, when a structure with rough interfaces is considered, the enforcement of the original non uniform boundary conditions through the stratification (m=0,..., N-1) can be addressed by writing down a linear system of equations with the aid of the matrix formalism (47)-(48) with m=0,..., N-1. As a result, the formulation of non-uniform boundary conditions in matrix notation (47)-(48) enables a systematic method for solving the scattering problem: For the N-layer stratification of Figure 1, we have to find 2N unknown expansion coefficients, using N vectorial equations (47), i.e., 2N scalar equations. It should be noted that, for the considered configuration, the relevant scattering coefficients SN+q(1)(k⊥),S0−q(1)(k⊥) are obviously supposed to be zero. The scattering problem, therefore, results to be reduced to a formal solution of a linear equation system. We finally emphasize that here we are interested in the scattering from the stratification; therefore, the determination of the unknown expansion coefficients S0+q(1)(k⊥) of the scattered wave into the upper half-space is required only. Full expressions for these coefficients are reported in (Imperatore et al. 2009a).

6. BPT Closed-form Solution

The field scattered upward in the upper half-space in the first-order limit can be written in the form (see (42)-(45)):

By employing the method of stationary phase, we evaluate the integral (51) in the far field zone, obtaining:

with q {v, h} is the polarization of the scattered field. The scattering cross section of a generic (nth) rough interface embedded in the layered structure can be then defined as

where < > denotes ensemble averaging, where q {v, h} and p {v, h} denote, respectively, the polarization of scattered field and the polarization of incident field, and where A is the illuminated surface area. The estimate of the mean power density can be obtained by averaging over an ensemble of statistically identical interfaces. Therefore, considering that the (spatial) power spectral density Wn(κ) of nth corrugated interface is defined as in (32), the scattering cross section relative to the contribution of the nth corrugated interface, according to the formalism used in [Franceschetti et al. 2008], can be expressed as:

with p, q {v, h} denoting, respectively, the incident and the scattered polarization states, which may stand for horizontal polarization (h) or vertical polarization (v); the coefficients α˜qpm,m+1 are relative to the p-polarized incident wave impinging on the structure from upper half space 0 and to the q-polarized scattering contribution from structure into the upper half space, originated from the rough interface between the layers m, m+1:

where ℑm|0p(k⊥) are the generalized transmission coefficients in upward direction (see (11)). Furthermore, we stress when the backscattering case ( k^⊥s×k^⊥i=0 ) is concerned, our cross-polarized scattering coefficients (55)-(58) evaluated in the plane of incidence vanish, in full accordance with the classical first-order SPM method for a rough surface between two different media (Ulaby et al, 1982) (Tsang et al., 1985).

We now show that the solution, given by the expression (55)-(58), is susceptible of a straightforward generalization to the case of arbitrary stratification with N-rough boundaries. Taking into account the contribution of each nth corrugated interface, the global bi-static scattering cross section of the N-rough interface layered media can be expressed as:

with p, q {v, h}, where the asterisk denotes the complex conjugated, where α˜qpi,i+1 are given by (55)-(58), and where the cross power spectral density Wij , between the interfaces i and j, for the spatial frequencies of the roughness is given by (33). As a result, the scattering from the rough layered media is sensitive to the correlation between rough profiles of different interfaces. In fact, a real layered structure will have interfaces cross-correlation somewhere between two limiting situations: perfectly correlated and uncorrelated roughness. Consequently, the degree of correlation affects the phase relation between the fields scattered from each rough interface. Obviously, when the interfaces are supposed to be uncorrelated, the second term in (59) vanishes and, in the first-order approximation, the total scattering from the structure arises from the incoherent superposition of radiation scattered from each interface.

As a result, an elegant closed form solution is established, which takes into account parametrically the dependence of scattering properties on structure (geometric and electromagnetic) parameters. In addition, as it will be shown in the following, the proposed global solution turns out to be completely interpretable with basic physical concepts, clearly discerning the physics of the involved scattering mechanisms.

7. Generalized Scattering Matrix

In this section, to emphasize the polarimetric character of the BPT solution, we introduce the generalized bistatic scattering matrix of the layered rough media, which can be then formally expressed by:

In particular,

characterizes the polarimetric response of the generic (mth) rough interface of the layered structure, for a plane wave incident in direction ki and for a given observation direction ks , with

and wherein

where we have introduced the notation

It should be noted that (63)-(67) are obtained directly by (55)-(58) by making use of (11). Note also that the coefficients α˜qpm,m+1 are expressible in a direct closed-form and depend parametrically on the unperturbed structure parameters. We also emphasize that the scattering configuration we have considered is compliant with the classical Forward Scattering Alignment (FSA) convention adopted in radar polarimetry.

Denoting with the superscript T the transpose, it can be verified that the scattering matrix satisfies the following relationship (Imperatore et al. 2009b)

which concisely expresses the reciprocity principle of the electromagnetic theory, as expected. This is to say that the scattering experiment is invariant for interchanging the role of transmitter and receiver. Note that the inversion of the projections on the z=0 plane k⊥i and k⊥s are directly related to the inversion of the incident and scattered vector wave ki=k⊥i−kz0iz^ and ks=k⊥s+kz0sz^ , respectively.

As a result, the presented closed-form solution permits the polarimetric evaluation of the scattering for a bi-static configuration, once the three-dimensional layered structure’s parameters (shape of the roughness spectra, layers thickness and complex permittivities), the incident field parameters (frequency, polarization and direction) and the observation direction are been specified. Therefore, our formulation leads to a direct functional dependence (no integral evaluation is required) and, subsequently, allows us to show that the scattered field can be parametrically evaluated considering a set of parameters: some of them refer to an unperturbed structure configuration, i.e. intrinsically the physical parameters of the smooth boundary structure, and others which are determined exclusively by (random) deviations of the corrugated boundaries from their reference position. Procedurally, once the generalized reflection/transmission coefficients are recursively evaluated, the (63-67) can be than directly computed, so that the scattering cross section (59) or the generalized scattering matrix (60) of a structure with rough interfaces can be finally predicted. Finally, it should be noted that the method to be applied needs only the classical gently-roughness assumption, without any further approximation.

8. Unifying Perspective on Perturbation Approaches