Remote sensing of layered random media using the radiative transfer theory

Authors

  • Saba Mudaliar

    Corresponding author
    1. Sensors Directorate, Air Force Research Laboratory, Wright-Patterson AFB, Ohio, USA
    • Corresponding author: S. Mudaliar, Sensors Directorate, Air Force Research Laboratory, Wright-Patterson AFB, 2241 Avionics Circle, Dayton, OH 45433, USA. (saba.mudaliar@wpafb.af.mil)

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Abstract

[1] The radiative transfer (RT) approach is widely used in remote sensing applications. Although this approach involves approximations, they are often not explicitly stated or explained. The RT approach for random media with nonscattering boundaries has been well studied, and the underlying assumptions are clearly documented. In contrast, our problem has scattering boundaries which are randomly rough. In order to better understand the RT approach to our problem, we adopt a statistical wave approach for modeling multiple scattering in the combined problem of random media and rough surfaces. The geometry of our problem consists of a multilayer discrete random medium with rough boundaries which are planar on the average. The statistical characteristics of the random medium in each layer are independent of each other and independent of the statistics describing the rough interfaces. Using the Green's functions of the problem without the volumetric fluctuations, we represent our problem as a system of integral equations. Employing the T-matrix description, we first average with respect to volumetric fluctuations to obtain a system of integral equations. We next average with respect to surface fluctuations, apply the weak surface correlation approximation, and arrive at a closed system of integral equations for the first and second moments of the electric fields. We use the Wigner transforms to relate the coherence functions to radiant intensities. On applying the quasi-uniform field approximation, we hence arrive at a system of equations identical to those used in the RT approach. In this process, we find that there are more conditions involved in the RT approach to our problem than widely believed to be sufficient. The important additional conditions are the following: (a) the thickness of layers are larger than the mean free path of the layer medium, and (b) the character of interface roughness is such that weak surface correlation approximation is applicable.

1 Introduction

[2] The radiative transfer (RT) theory is widely used in remote sensing problems [Lenoble, 1993; Thomas and Stamnes, 1999; Natsuyama et al., 1998; Liang, 2003; Arsar, 1989; Ulaby et al., 1986; Jin et al., 2007]. Often, the model of layered random medium with rough interfaces is used. Multiple scattering processes in this structure are represented by the RT equations. Although quite successful in numerous applications in various disciplines, it is known that the RT approach involves approximations. Often people in the remote sensing community are not quite familiar with the approximations involved in the RT approach, and hence, there has been inappropriate use of the RT approach in the literature. Since the phenomenological RT theory [Chandrasekar, 1960; Sobolev, 1963] was first developed for light scattering in planetary atmospheres, the RT conditions prevalent in the atmospheric context have been popularly identified as sufficient conditions for employing the RT theory. However, we notice that the RT theory has also been freely used for a variety of different problems with complex geometries [Ulaby et al., 1990; Mobley, 1994; Saatdjian, 2000; Modest, 1993; Sato and Fehler, 1998]. It is not clear whether the classical conditions associated with the RT theory are sufficient in all situations. In this paper, we will review the approximations involved and clarify misconceptions. In order to better understand the RT approach, we employ the more rigorous statistical wave theory to the problem and hence make the transition to the RT equations. In this process, we clarify and explain the assumptions or approximations involved in the RT approach. By following this procedure, we found that there are more conditions embedded in the RT approach than widely believed to be sufficient. Two important additional conditions required are the following: (a) the thickness of the layers are of the same order or larger than the mean free path of the scattering media, and (b) surface roughness of interfaces are such that the weak surface correlation approximation is applicable. For our study, we have considered a multilayer random medium composed of discrete scatterers (see Figure 1). By considering several special cases of this general problem, we show that the number of conditions implied in the RT approach reduces with simpler geometries. These conclusions are not just for the case of discrete random media. Similar conditions apply for other models where the volumetric fluctuations were modeled as a random continuum [Mudaliar, 2010]. The random continuum concept is rather abstract and unsuitable for modeling several problems encountered in applications. Perhaps, the most serious drawback of the continuum model is that it is often impossible to relate the key statistical parameters to observable quantities. The current paper is devoted to the remote sensing applications. A discrete scatterer model is most appropriate and realistic for modeling real world problems [Mishchenko et al., 2006; Liang et al., 2005; Fung and Chen, 1981; Lam and Ishimaru, 1993; Ulaby et al., 1990; Olsen et al., 1976; Ito and Ogichi, 1987; Nghiem et al., 1993]. Notice that the analysis used in the discrete random media is quite different from the continuum case [Foldy, 1945; Lax, 1951; Twersky, 1964; Tsang et al., 1984; Lang, 1981]. Furthermore, the conditions and assumptions embedded in the RT approach are also different in the discrete model case.

[3] The paper is organized as follows. First, we describe the geometry of the problem. Next, we give the radiative transfer approach to the problem. The next section is on the statistical wave approach to the problem. This occupies the major part of the paper. It deals with the derivation and analysis of the first and second moments of the wave functions. A transition is next made to RT equations. Next, we have a detailed discussion of all the results obtained in this paper. In particular, we carry out a comparative study of the conditions embedded in the RT approach to layered random medium with scattering rough interfaces and the random medium problem with nonscattering boundaries.

2 Description of the Problem

[4] The geometry of the problem is shown in Figure 1. We have an N-layer random medium stack with rough interfaces which, on the average, are parallel planes. Let εj be the permittivity of the background medium, and let εjs be the permittivity of the scatterers in the jth layer. The location and orientation of the scatterers are random functions characterizing the fluctuations. On the average, the problem is translationally invariant and isotropic in the azimuth. We assume that the volumetric fluctuations in each layer are statistically independent of each other. Let Nj be the number of scatterers, and let ρj be the density of the scatterers (number of scatterers per unit volume) of the jth layer. The permeability of all the layers is that of free space. The rough interfaces are described as z=zj+ζj(r), where zj denote the location of the unperturbed interfaces. The ζj's are the zero mean isotropic stationary random processes independent of volumetric fluctuations of the problem. Let z0=0, and let dj be the thickness of the jth layer. The media above and below the stack are homogeneous with parameters ε0, k0, and εN+1, kN+1, respectively. This system is excited by a monochromatic electromagnetic plane wave from above, and we are interested in modeling the resulting multiple scattering process.

Figure 1.

Geometry of the problem.

3 Radiative Transfer Approach

[5] Multiple scattering process in a complex environment is well described by the radiative transfer theory. This theory is not only conceptually simple but also very efficient. The fundamental quantity here is the specific intensity I which is governed by the following equation [Chandrasekar, 1960; Sobolev, 1963; Ishimaru, 1997]

display math(1)

The specific intensity I is a phase-space quantity at position rand direction inline image. inline image is the extinction matrix which is a measure of loss of energy due to scattering in other directions. inline image is the phase matrix representing increase in energy density due to scattering from neighboring elements. Ω is the solid angle subtended by inline image. Given the statistical characteristics of the medium, one can calculate the phase matrix using the single scattering theory for elements that constitute the random medium of the layer [van de Hulst, 1981; Kerker, 1988; Bohren and Huffman, 2004; Mishchenko et al., 2002]. The extinction matrix is hence calculated using the optical theorem. The specific intensity in each layer is governed by an equation similar to (1). Since our layer problem has translational invariance in azimuth, the RT equation for the mth layer takes the following form,

display math(2)

where the subscript m is used to indicate that the quantity corresponds to that of the mth layer and θ is the elevation angle associated with inline image. This set of RT equations is complemented by a set of boundary conditions. On the mth interface, we have

display math(3)

The boundary conditions on the (m−1)th interface are given as

display math(4)

where inline image and inline image are the local reflection and transmission Mueller matrices. To be more specific, inline imagerepresents the reflection matrix of waves incident from medium n on the interface separating medium m and medium n. The superscripts u and d indicate whether the intensity corresponds to a wave traveling upward or downward. These Mueller matrices are often calculated using some asymptotic theory such as the Kirchhoff approximation [Voronovich, 1999; Beckmann and Spizzichino, 1987; Ulaby et al., 1986; Tsang et al., 1985]. The integration in these expressions are over a solid angle (hemisphere) corresponding to inline image. For the time-harmonic electromagnetic plane wave incident on this stack from above the downward traveling intensity in Region 0 is given as

display math(5)

where B0 is the intensity of the incident plane wave and {θi,φi} describes its direction. Since there is no source or scatterer in Region N+1,

display math

We should point out that the radiative transfer approach as applied to a particular problem is only a model based on certain assumptions. Since the RT theory is used in a variety of applications, the particular assumptions involved are described in terms of different terminologies, specific to the discipline where it is used. One good way to understand in more general terms the RT approach and the underlying assumptions is to compare it with the more rigorous wave approach. For the case of an unbounded random medium, this kind of study was carried out in the 1970s [Barabanenkov et al., 1972; Barabanenkov, 1975]. From that study, we learn that the radiative transfer theory can be applied under the following conditions:

  1. [6] Quasi-stationary field approximation.

  2. [7] Sparse distribution.

  3. [8] Statistically uniform distribution of scatterers.

[9] These are the basic conditions that we associate with the RT approach. However, our problem has bounded structures with rough interfaces. The question is this: are the above conditions sufficient to apply the RT approach for our problem? This is the motivation for this paper. We follow the statistical wave approach to this problem, derive the equations for the intensities, and hence make the transition to the RT equations. This procedure enables us to better understand the conditions for using the RT approach for our problem.

4 Statistical Wave Approach

[10] Electromagnetic waves in the layered structure are governed by the following system of equations:

display math(6)

where

display math(7a)
display math(7b)
display math(7c)

Vji is the domain of ith scatterer and Nj is the number of scatterers in layer j. Thus, vj represents the volumetric fluctuations in Region j. For the homogeneous regions above and below the stack, we have the following corresponding wave equations:

display math(8a)
display math(8b)

The boundary conditions at the jth interface are as follows

display math(9a)
display math(9b)

where inline image is the unit vector normal to the jth interface with normal pointing into medium j. This system is complemented by the radiation conditions well away from the stack. We assume that we know the solution to the problem without volumetric fluctuations and denote it as inline image. The Green's functions inline image for this problem are governed by the following set of equations:

display math(10a)
display math(10b)
display math(10c)

where inline image here is the unit dyad. The boundary conditions given above are for the jth interface. Similarly, on the (j−1)th interface, we have the following boundary conditions:

display math(11a)
display math(11b)

Using these Green's functions and the radiation conditions, the wave functions for our combined problem can be represented as

display math(12)

where Ωk={r⟂′;ζk<z<ζk−1}. Note that v0=vN+1=0. In order to carry out multiple scattering analysis with a distribution of discrete scatterers, it is convenient to employ the concept of transition operator inline image [Lax, 1951; Waterman, 1965; Varadan and Varadan, 1980]. Suppose we know the electric field El incident on the lth scatterer. We introduce the transition operator inline imagesuch that the electric field scattered by the lth scatterer is given as inline image. Using this concept (12) may be expressed as

display math(13)

Note that inline image depends only on the lth scatterer. To proceed further with the multiple scattering analysis, it is expedient to use symbolic representation of (13).

display math(14)

First, we average (14) with respect to volumetric fluctuations to get

display math(15)

where the subscript v denotes the volumetric averaging. Since there are Nk scatterers in the kth layer,

display math(16)

where p is the joint probability density function of finding the scatterers at inline image with orientations inline image. We assume that the positions and orientations are independent of each other. In other words,

display math(17)

Furthermore, assume that the orientation of the particle at position r1 is independent of the orientation of all other particles, which means

display math(18)

We next express the joint position probability density function as

display math(19)

where inline image is the conditional pdf of finding scatterers at inline image by fixing the lth scatterer at rl. The prime in rl′ denotes that rl should be omitted in the argument list of the conditional probability density function. Substituting this relation in (16), we obtain

display math(20)

where inline image denotes the conditional average with scatterer l fixed at rl. If Nk is large and the distance between scatterers is large, then we can make the following approximation

display math(21)

This is called the Foldy's approximation and is applicable for sparse media. Under this approximation, (15) becomes

display math(22)

where ρk is the density of the scatterers in layer k. Now operate the above equation by inline image to obtain

display math(23)

Next, average this over surface fluctuations to get

display math(24)

Note that the element transition operators are independent of surface fluctuations. From this, we see that

display math(25)

which means that the coherent propagation constants in the regions above and below the layered stack are unaffected by the fluctuations of the problem. However, they indeed get modified in the layered stack region. On writing (24) as

display math(26)

we infer that inline image represents the mean propagation constant, in operator form, for coherent waves in layer j.

[11] Since the problem is statistically homogeneous in azimuth, the mean fields in our system have the following form:

display math(27)
display math(28)

and

display math(29)

where the superscript p stands for the polarization, either horizontal or vertical. Note that inline image; the superscript p denotes the two polarizations. p is the unit vector representing polarization. The subscript i is used to indicate that the wave vector is in the incident direction. R and X denote respectively the mean reflection and transmission coefficients of the stack. Aj and Bj denote, respectively, the mean coefficients of upgoing and downgoing waves in the jth layer.

[12] Based on this, we can formulate the waves averaged with respect to volumetric fluctuations as

display math(30)
display math(31)

and

display math(32)

where Aj, Bj, R, and X are now the integral operators representing scattering from rough interfaces. The boundary conditions associated with the above equations at the jth interface are

display math(33)

and

display math(34)

The above system may be solved either numerically or by any of the asymptotic methods available in rough surface scattering theory [Beckmann and Spizzichino, 1987; Bass and Fuks, 1979; Voronovich, 1999] to evaluate the mean coefficients that appear in (27)(29).

[13] We proceed now to the analysis of the second moments, by starting with (12). For convenience, we write it in symbolic form as

display math(35)

We take the tensor product of this equation with its complex conjugate and average with respect to volumetric fluctuations to obtain

display math(36)

where K is the intensity operator of the volumetric fluctuations. Employing the weak fluctuation approximation, we approximate K by its leading term

display math(37)

[14] On substituting this in (36), we get

display math(38)

Next, we average (38) with respect to the surface fluctuations to get

display math(39)

where we have used the following approximation

display math(40)

We call this the “weak surface correlation” approximation, which we will find to be an important condition embedded in the RT approach to our problem.

[15] Equation (39) is the main equation representing multiple scattering process for our problem. One important goal for us is to investigate the conditions needed for employing the radiative transfer approach for our problem. With this in mind, we employ Wigner transforms. Note that (39) is an equation for the coherence function. On the other hand, the RT equation, as we saw earlier, is an equation for the specific intensity. Wigner transform serves as a bridge to link these two quantities [Yoshimori, 1998; Friberg, 1986; Marchand and Wolf, 1974; Pederson and Stamnes, 2000].

[16] We introduce Wigner transforms of waves and Green's functions as follows:

display math(41a)
display math(41b)
display math(42)

In terms of these transforms, (39) becomes

display math(43)

with

display math(44)

where inline image is the element transition operator in nth layer.

[17] The fact that our problem has translational invariance in azimuth implies the following:

display math(45a)
display math(45b)

Using these relations in (43), we obtain

display math(46)

where

display math(47)
display math(48)

inline image is the element scattering matrix in the nth layer. Since the medium is assumed to be sparse, interparticle scattering takes place in the far-field zone of each other. It is based on this fact that we have transitioned from inline image to inline image. Also, we have employed the on-shell approximation to inline image.

[18] To proceed further, we need to evaluate inline image. Note that we need to relate this system with that of RT, which involves the boundary conditions at the interfaces. Therefore, we need to identify the coherence functions corresponding to upgoing and downgoing wave functions. To facilitate this, we decompose inline image into its components,

display math(49)

where the first term is the singular part of the Green's function. The superscripts u and d indicate upgoing and downgoing elements of the waves. The other components are due to reflections from boundaries. These are formally constructed using the concept of surface scattering operators as follows [Voronovich, 1999],

display math(50)

where inline imageis the surface scattering operator. The superscripts a and b on inline image are used to indicate whether the waves are upgoing or downgoing. In the exponents, we have used the following convention: a,b=1 if the waves are upgoing; a,b=−1 if the waves are downgoing. The z component of the mean propagation constants in the nth layer is denoted as qn. Recall that inline image is the Wigner transform of inline image. The superscripts μ,ν stand for polarization, either h or v. When we use (42) to perform the Wigner transform, we ignore all cross terms. In other words, we make the following approximation,

display math(51)

where inline image is the Wigner transform of inline image.

[19] With the introduction of this representation for inline image in (43), we can trace upgoing and downgoing waves to obtain the following equations for the coherence function:

display math(52a)
display math(52b)

Note that summation over a,b={u,d} is implied in the above equations. The first term in these equations, inline image, represents the contribution due to surface scattering and has the following form:

display math(53)

where inline image is the amplitude of the upgoing wave in the mth layer after volumetric averaging is performed. This means that it is a random function of surface fluctuations. When we substitute (53) and the expressions for inline image in (52), we find that

display math(54)

On substituting this in (52) and differentiating with respect to z, we obtain the following transport equations:

display math(55a)
display math(55b)

where summation over a is implied. When the superscript a corresponds to u, the value of a in the argument of inline image is taken as +1; on the other hand, when the superscript a corresponds to d, the value of a in the argument of inline image is taken as −1. Since all quantities in (55) correspond to the same layer m, we have dropped the subscript m in inline image and inline image to avoid cumbersome notations. To obtain appropriate boundary conditions, we go back to the integral equation representations for inline imageand inline image, examine their behavior at the interfaces, and seek a relation between them. After considerable effort, we arrive at the following boundary conditions. At the (m−1)th interface, we have

display math(56)

with inline imagewhere inline image is the stack reflection matrix (not the local reflection matrix) for a wave incident from below on the (m−1)th interface. Similarly,

display math(57)

where inline image is the tensor product of stack reflection matrix for a wave incident from above on the (m−1)th interface. We were able to obtain these boundary conditions only after imposing certain approximations such as the one given below. Consider the following identity:

display math(58)

where inline image. Notice that this is an operator relation where all elements are operators. Taking the tensor product of (58) with its complex conjugate, we have

display math(59)

Next, we average (59) with respect to surface fluctuations and get

display math(60)

where we have approximated that the two tensor products in the middle are weakly correlated. A further approximation that we impose is given as follows

display math(61)

These are the kinds of approximations required to arrive at our boundary conditions.

5 Transition to Radiative Transfer

[20] We now transition from the transport equation (55) to the phenomenological radiative transfer equation discussed earlier. To accomplish this, we have to link the key quantities of waves and radiative transfer, viz., coherence function and specific intensity. The relation between them is obtained by computing the Poynting vector using the two concepts. One of the fundamental results of electromagnetic wave theory is the Poynting vector given as

display math

For time-harmonic transverse electromagnetic waves, this becomes

display math

From Wigner transform relations, we have

display math(62)

The average Poynting vector is also related to the specific intensity as

display math(63)

where I is the first element of the Stokes vector. The above two relations suggest that the following definition for the specific intensities

display math(64)

Now we can transition to the phenomenological RT equations. Using the relation between inline image and I, we change the integration variable to solid angle and arrive at the following equations,

display math(65a)
display math(65b)

where inline image is the extinction matrix and inline image is the phase matrix. Implicit summation over subscript j is assumed in (65). To facilitate comparison with the results of Ulaby et al. [1986] and Lam and Ishimaru [1993], we have used a modified version of Stokes vector [Ishimaru, 1997]. Instead of the standard form {I,Q,U,V}, we use {(I+Q)/2,(IQ)/2,U,V}. The subscript of I in (65) denotes the element number of our modified Stokes vector. Although the structure of this equation is identical to that of the RT (equation (2), the elements of the phase matrix and the extinction matrices are not the same. The primary reason is because of the differences in the real part of the mean propagation constants of horizontally and vertically polarized waves. On assuming that qh′=qv′=kmz′, we obtain the following expressions for the extinction and phase matrices:

display math(66)
display math(67)

where

display math(68a)
display math(68b)
display math(68c)
display math(68d)

We have suppressed the arguments for brevity. For instance,

display math(69)

f's are the elements of the scattering matrix inline image defined as follows:

display math

In the {h,v} basis the scattering matrix is given as

display math

Note that these transport equations (65) are identical to those of classical RT equations (2) that we described in section 2. Thanks to our statistical wave approach, we now have explicit expressions for the extinction matrix and phase matrix in terms of the statistical parameters of the problem. Let us now turn our attention to the boundary conditions (BC). In our wave approach, we obtained BCs in terms of “stack” reflection matrix inline image, whereas in the RT approach, the BCs are given in terms of the local interface reflection matrices. We can readily reconcile with this apparent difference. Note that the BC in the wave approach forms a closed system whereas in the RT approach, it is open (linked to adjacent layer intensities). Let us take a look at the BC at the (m−1)th interface. inline image can be expressed in terms of inline imageas follows,

display math(70)

This is the relation between the stack reflection coefficients of adjacent interfaces. inline image and inline imageare the local (single interface) reflection and transmission matrices at the (m−1)th interface. On operating inline image with (70), we get

display math(71)

Notice that this boundary condition now involves only local interface Fresnel coefficients. Take the tensor product of (71) with its complex conjugate and average with respect to surface fluctuations. Employing the Wigner transform operator on this, we obtain a boundary condition at the (m−1)th interface similar to that of the RT system (see (4)). However, the reflection and transmission matrices used in the RT system correspond to unperturbed medium as opposed to the average medium as in the case of the wave approach.

[21] Similarly, we write inline imagein terms of inline imageand hence obtain the BC at the mth interface as

display math(72)

Take the tensor product of (72) with its complex conjugate and average with respect to surface fluctuations. Employing the Wigner transform operator on this, we obtain boundary condition at the mth interface identical to (3) (after making the approximations as before).

6 Discussion

[22] The main goal of this paper is to critically examine the radiative transfer (RT) approach to remote sensing of layered random medium with rough interfaces. Such problems are often encountered in remote sensing applications. Several assumptions are embedded in the radiative transfer approach to this problem. There are numerous works in the literature on the study of radiative transfer theory and the underlying assumptions. However, all have dealt with unbounded geometry or bounded geometry with nonscattering boundaries. Our interest in this paper is on the problem of layered random medium with irregular scattering boundaries. There does not exist any critical study of RT approach to this kind of geometry. Our study has shown that there are additional conditions embedded in the RT approach to this problem beyond those for the problem with unbounded or nonscattering geometries. The additional conditions are imposed on the thickness of the layers and on the roughness of the interfaces. These facts are not well known to the users of the RT approach. One purpose of this paper is to inform the remote sensing community about these additional conditions so that they have a good idea on when the RT approach is acceptable for the application at hand.

[23] To enable the critical study of the RT approach, we developed a statistical wave approach for the combined problem of layered random medium with rough interfaces. Such a foundation is essential for this study because there is no suitable statistical wave theory in the literature that is appropriate for multiple scattering analysis for our problem. From this study, we find that the coherent waves in our problem behave like waves in a layered homogeneous medium with planar interfaces. The propagation constants of this layered medium are primarily determined by the statistical properties of the local medium. They are weakly dependent on interface roughness. Their dependence on the medium fluctuations of layers other than the one under consideration is of higher order. In contrast, the effective Fresnel coefficients are influenced by not only statistical properties of the local interface but also on the statistical properties of the adjacent media. Their dependence on the surface roughness of other interfaces and media of other layers are of higher order. We notice that diffuse scattering process in our problem is fairly complicated because of volume-surface interactions. In recent years, there has been a proposition that the combined problem of volumetric scattering and surface scattering be split into two parts: one due to volumetric scattering and the other due to surface scattering. The surface scattering part is that of the layered structure with rough interfaces and homogeneous media with effective permittivities. The volumetric scattering part is that from the layered random medium with unperturbed flat interfaces. It is found from our study that this kind of splitting is not, in general, possible unless the fluctuations of the problem are very weak.

[24] The results obtained in this paper apply to two situations. Below, we consider them separately.

  1. [25] Time-varying problem: Here the parameters of the problem vary with time. However, an important assumption must be made to simplify the analysis. The time constant associated with the parameter fluctuations of the problem should be much larger than the time constant associated source signal. The time average is taken over a period much larger than the time constant associated with the medium fluctuations (that includes volumetric and surface fluctuations). Now we impose the ergodicity hypothesis and equate time averages to ensemble averages. One such application where this situation occurs is in electromagnetic wave propagation and scattering in atmosphere.

  2. [26] Time independent problem: Here all the parameters are independent of time. However, they undergo spatial variations. The statistical aspects of the problem enter through spatial fluctuations. Therefore, all averages are ensemble averages. Obviously, then the question of ergodicity does not arise here.

[27] The transport equations as derived in (55) along with the boundary conditions (56) and (57) are important results of the paper. This system describes the behavior of coherence functions of upward and downward traveling waves in each layer of the problem. Several important physical quantities can be directly calculated using these coherence functions.

  • 1.Poynting-Stokes tensor: The Poynting-Stokes tensor [Mishchenko, 2010, 2011], a key quantity in radiative transfer, is related to our coherence function as follows:
    display math(73)
    where η is the intrinsic impedance of the medium where z is located. inline image and inline image are the unit propagation vectors associated with inline imageand inline image, respectively. Ω+ and Ω indicate that the domain of integration in the upper and lower hemisphere, respectively.
  • 2.Specific intensity: We showed in this paper how the phenomenological radiative transfer equation (65) may be derived from our equation for coherence function by making the following link between specific intensity and coherence function:
    display math(74)
  • 3.Average Poynting vector: This quantity which is vital for radiation budget computation [Thomas and Stamnes, 1999; Liou, 1992] is derived from the coherence function as follows:
    display math(75)
  • 4.Bistatic scattering coefficient: In our problem, the system is excited by a plane wave incident on the zeroth interface from above. The bistatic scattering coefficient, σr in Region 0 (in the direction corresponding to k) is given as
    display math(76)
    where inline image. Suppose the incident plane wave is horizontally polarized. Then, the hh component of left-hand side (LHS) represents inline image; the vv component represents inline image. Similarly, if the incident wave is vertically polarized, the hh component of LHS represents inline image; the vv component represents inline image. We have used the superscript r to indicate that this quantity is a reflection type scattering coefficient since the source and observation points are located in the same Region 0. This kind of scattering coefficient is extensively used in remote sensing applications [Ulaby et al., 1986; Elachi and van Zyl, 2006; Arsar, 1989; Tsang et al., 1985].

[32] Next, suppose that the observation point is in Region N+1 while the source is still in Region 0. Now the bistatic scattering coefficient is given as

display math(77)

We call this the transmission scattering coefficient and indicate it with the superscript t. This quantity is quite useful in a variety of imaging applications.

  • 5.Passive remote sensing: The key quantity of interest in passive remote sensing is emissivity [Elachi and van Zyl, 2006; Ulaby et al., 1986; Arsar, 1989] e given as
    display math(78)
    where rpc and rpi are the coherent and incoherent reflectivities defined as shown below.
    display math(79)
    display math(80)
    where inline image is the qpth element of the stack reflection coefficient inline image. inline image is the reflection scattering coefficient defined in (74).

[34] Indeed, there are other useful physical quantities [Elachi and van Zyl, 2006; Liou, 2001; Kokhanovsky, 2004] that can be obtained from the coherence function derived in (55). We have just given some examples relevant to remote sensing problems.

[35] In order to understand the foundations of the radiative transfer approach, we made the transition from the governing equation for coherence function to the phenomenological radiative transfer equation. We made the transition from statistical wave theory to radiative transfer theory by employing the Wigner transform and computing the average Poynting vector by using wave theoretical methods and transport theoretical concepts.

[36] Having made this transition, it is now instructive to itemize the assumptions. The three basic conditions required are the following:

  • 1.Quasi-stationary field approximation.
  • 2.Sparse distribution of scatterers.
  • 3.The number of particles in each layer is large.

[40] These are the basic conditions necessary for the unbounded random medium problem. However, if the medium is bounded, we need to impose the following additional conditions:

  • 4.Layer thickness must be of the same order or greater than the mean free path of the layer.

[42] When the interfaces are randomly rough, we further need the following conditions:

  • 5.Weak surface correlation approximation.
  • 6.All fluctuations of the problem are statistically independent of each other.

[45] The above are the essential conditions associated with the RT approach. There are several secondary assumptions that we have employed for simplifying the analysis and discussion. They are the following:

  1. [46] The location and orientation are the only randomly varying aspects of the media.

  2. [47] All particles in a particular layer are of the same type and are uniformly distributed. However, they can be of different type in different layers.

  3. [48] The problem is translationally invariant and isotropic in the azimuth.

  4. [49] The problem is assumed to be time independent.

  5. [50] If the problem happens to be time-varying, the following additional conditions need to be imposed: (a) The time constant of statistical parameters is much longer than that of the signal, and (b) the problem is ergodic.

  6. [51] The position and orientation of each particle are statistically independent of each other and of all other particles.

  7. [52] The interfaces of the random medium are parallel planes on the average.

[53] Let us make some remarks on the secondary assumptions.

[54] Assumption 1: One can add to this other variations such as size, shape, and material properties at the price of more complicated analysis.

[55] Assumption 2: All particles need not be of the same type. We can have multispecies in each layer. The distributions can be arbitrary. It is not necessary for them to be uniform. However, such details will add to the complexity of the results.

[56] Assumption 3: While translational invariance is an important assumption for our analysis, the statistical fluctuations need not be isotropic in azimuth. Layered random media without translational invariance is a very complex problem not considered in this paper. We can indeed take into account anisotropy in statistical fluctuations. However, we will have a tensor form of effective permittivity and associated mode patterns.

[57] Assumption 4: Time independence is another assumption intended to simplify the analysis.

[58] Assumption 5: The results obtained in the paper apply to time-varying case provided that the additional assumptions are imposed.

[59] Assumption 6: This is not a fundamental assumption. This property may be derived from “sparse medium.”

[60] Assumption 7: This is an important assumption necessary for our analysis. However, this assumption is not essential for the main character of our results.

[61] We would like to make a few more remarks before ending the discussion on our assumptions: (a) In RT theory, the medium is assumed to be sparse, and hence, the “refraction effects” of the fluctuations are ignored. Thus, in the boundary conditions, we should use the background medium parameters rather than the effective medium parameters as derived in our statistical wave theory. (b) To arrive at (65), we have ignored the contribution of evanescent modes. (c) We found that the extinction coefficients calculated in the wave approach and the RT approach are different and only after further approximations can they be made to agree with each other.

[62] Mishchenko et al. [2006] (hereinafter referred to as MTL for brevity) derived the vector radiative transfer equation (VRTE) for a bounded discrete random medium using a rigorous microphysical approach. This enabled them to identify the following assumptions embedded in the VRTE.

  1. [63] Scattering medium is illuminated by a plane wave.

  2. [64] Each particle is located in the far-field zone of all other particles, and the observation point is also located in the far-field zones of all the particles forming the scattering medium.

  3. [65] Neglect all scattering paths going through a particle two or more times (Twersky approximation).

  4. [66] Assume that the scattering system is ergodic and averaging over time can be replaced by averaging over particle positions and states.

  5. [67] Assume that (i) the position and state of each particle are statistically independent of each other and those of all other particles and (ii) spatial distribution of the particles throughout the medium is random and statistically uniform.

  6. [68] Assume that the scattering medium is convex.

  7. [69] Assume that the number of particles N forming the scattering medium is very large.

  8. [70] Ignore all the diagrams with crossing connections in the diagrammatic expansion of the coherency dyadic.

[71] First, notice that MTL dealt with the problem with nonscattering boundary. For this problem, only the first three of our essential conditions and the first six of our secondary conditions should apply. The list of assumptions given by MTL consists of a combination of both essential and secondary conditions. Furthermore, the conditions that MTL obtained are not identical with the ones that we derived even when restricting attention to the random medium problem with nonscattering boundary. This is because the methods that we employed are different from theirs. Here below, we address the differences and provide explanation for the discrepancies.

[72] MTL 2: MTL have made deliberate use of the far-field approximation. We have not explicitly used this approximation. Instead, we have used the more general quasi-uniform field approximation. To illustrate this, consider electric fields at two points r1 and r2:

display math

The coherence function of these fields is given as

display math

where inline image, inline image, and r=r1r2. In the quasi-uniform field approximation, we assume that 〈E(r1)⊗E(r2)〉 weakly varies with R and strongly varies with r. In contrast, MTL employed the far-field approximation under which inline image is independent of position vectors. For our problem of layered random medium, it is unrealistic to explicitly employ the far-field approximation.

[73] MTL 3: We have not explicitly used the Twersky approximation. Instead, we made use of the fact that the number of particles is large and hence approximated conditional averages by unconditional averages. Incidentally, we arrive at the same result as that of MTL for the unbounded random medium problem.

[74] MTL 6: In our problem, we have distinct scattering boundaries and the character of the waves exiting or entering them are explicitly contained in the boundary conditions. Hence, convexity of the scattering medium is not a necessary condition for us.

[75] MTL 8: In our approach, we used the weak fluctuation criterion and retained only the leading term of the intensity operator. Notice that the ladder term is the leading term of our intensity opeartor. Crossed terms are higher order terms which have contribution only in the backscattering direction. Hence, such terms do not appear in our results.

[76] Let us reiterate that MTL with whom we made detailed comparison have considered a bounded random medium with nonscattering boundary. Ours is a significantly more complicated problem because of the layered structure and scattering interfaces with irregular geometry. Thus, our problem contains volume-surface interactions because of multiple scattering. Nobody has critically examined the RT approach to this problem. By following a systematic and rigorous approach, we found that the RT approach to our problem requires additional conditions beyond those sufficient for the problem with nonscattering boundaries. It is expected that some of our conditions are equivalent to those of MTL. The key results of the paper lie in the extra conditions. These additional conditions that are special to our geometry are the following: (a) The layer thickness must be of the same order or larger than the mean free path of the medium and (b) weak surface correlation approximation. (c) All volumetric fluctuations and surface fluctuations are independent of each other. These are the essential extra conditions. Following are some secondary extra conditions that we have imposed to arrive at our results. The surface fluctuations are the following: (a) translationally invariant and isotropic in azimuth, (b) statistically homogeneous, and (c) single-valued. These assumptions were used to simplify the analysis and discussion.

[77] The existence of these extra conditions means that RT approach, as it is popularly conceived, may be applied to only a limited type of layered random media. In other words, the accuracy which can be obtained using RT theory depends on the geometry of the problem. For certain geometries such as the one discussed in this paper, the results obtained using RT approach can be grossly inaccurate. We have obtained these results using a systematic analysis of the macroscopic Maxwell's equations. Nobody else has observed that additional conditions are involved in the RT approach to layer geometry beyond those for the unbounded geometry of with nonscattering boundaries. In view of these remarks, the results of this paper are important to the remote sensing community. Until now, RT theory has been taken as a fundamental law and applied to variety of random media with complex geometries. This paper shows that under what conditions RT theory may be used to layered random media.

[78] To summarize, we have inquired into the assumptions involved in adopting the radiative transfer approach to scattering from layered random medium with rough interfaces. To facilitate this inquiry, we adopted a wave approach to this problem and derived the governing equations for the first and second moments of the wavefields. We employed Wigner transforms and transitioned to the system corresponding to that of radiative transfer approach. In this process, we found that there are more conditions implicitly involved in the RT approach to this problem than it is widely believed to be sufficient. These additional conditions are placed on the thickness of the layers and on the roughness of the interfaces. With the recent development of fast and efficient algorithms for scattering computations and the enormous increase in computer resources, it is now feasible to take an entirely numerical approach to this problem without imposing any approximations. In spite of such developments, to keep the size of the problem manageable, only special cases have been studied thus far [Giovannini et al., 1998; Pelosi and Coccioli, 1997; Pak et al., 1993; Sarabandi et al., 1996]. Hence, it is very much of relevance, interest, and convenience to apply the RT approach to these problems. However, one should keep in mind the assumptions involved in such an approach. Otherwise, interpretations of results based on RT theory can be misleading.

Acknowledgment

[79] The author thanks the U.S. Air Force Office of Scientific Research for support.

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