## 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.