## 1. Introduction

[2] Electromagnetic wave scattering from rough layers (i.e. two rough interfaces separating three homogeneous media) has been investigated in many recent papers. While some previous studies [*Kuo and Moghaddam*, 2006; *Duan and Moghaddam*, 2010] have utilized fully numerical solutions of the boundary value problem, the computational expense for such models is high, limiting their practical application. The much lower computational expense of approximate methods makes their use desirable, so long as their inherent approximations are satisfied. Recent work developing approximate methods for the rough layer problem include *Pinel et al.* [2007, and references therein], as well as *Kuo and Moghaddam* [2007], *Demir* [2007], *Berginc and Bourrely* [2007], and *Imperatore et al.* [2010]. The majority of these studies utilize a small perturbation solution [*Yarovoy et al.*, 2000; *Tabatabaeenejad and Moghaddam*, 2006; *Kuo and Moghaddam*, 2007; *Imperatore et al.*, 2010; M. A. Demir et al., A study of the fourth-order small perturbation method for scattering from two-layer rough surfaces, submitted to*IEEE Transactions on Geoscience and Remote Sensing*, 2011], which assumes that the heights of the interfaces are small compared to the electromagnetic wavelength. While this is most often the case of interest for radar remote sensing, since it is at low frequencies that sub-surface interfaces would be most likely to be observable, it is nevertheless of interest to develop models applicable to surfaces with roughness heights that are moderate to large in terms of the electromagnetic wavelength so that returns either at higher frequencies or for rougher surfaces at lower frequencies can be investigated.

[3] An appropriate approach for this situation is the Kirchhoff-tangent plane approximation (KA), which in conjunction with its high-frequency analytical solving method (i.e., the geometric optics approximation) was extended to predict scattering from rough layers for two-dimensional (2D) problems first [*Pinel et al.*, 2007; *Pinel and Bourlier*, 2008], and then extended to three-dimensional (3D) problems for flat [*Pinel et al.*, 2009] and rough [*Pinel et al.*, 2010] lower interfaces.

[4] This method obtains a mathematical expression for scattered fields by iterating the KA for each scattering at a rough interface inside the rough layer. While the initial formulation produces numerous integrals that must be evaluated numerically, the method of stationary phase (MSP) reduces the number of numerical integrations by including only specular reflection processes. A further application of the geometric optics (GO) approximation, which assumes that only closely located correlated surface points contribute to the NRCS, further reduces the computational complexity. In general the model is applicable only for surfaces with heights that are moderate to large in terms of the electromagnetic wavelength, such that the coherent component of the fields is negligible in comparison with their incoherent components.

[5] In section 2, the NRCS under the GO-layer model is reviewed, and the extension of the model to predict the polarimetric covariance of scattered fields is presented. Such an extension is motivated by interest in the potential benefits of polarimetric measurements for remote sensing, including those for near-specular bistatic measurements as in the sensing of land surfaces with Global Navigation Satellite System (GNSS) reflectivity [*Zavorotny et al.*, 2010]. Sample geoscience and remote sensing applications are presented in section 3, including examination of fully polarimetric returns in the complete hemispherical bistatic scattering pattern. It is shown that some portions of the bistatic pattern provide enhanced visibility of sub-surface returns, so that utilization of these portions may be of interest in future measurements. A renewed examination of a problem that has received extensive previous interest, the remote sensing of sub-surface properties in arid regions, is also performed in order to assess the utilization of higher microwave frequencies. Particular angular regions for monostatic observations of sub-surface properties are then determined as a function of surface roughness and layer dielectric properties.

[6] In general the results provide useful insight into the impact of sub-surface layers of radar remote sensing of land surface properties, and can be utilized in the design of future sensors for sub-surface sensing applications.