## 1. Introduction

[2] The method of moments (MoM) is a powerful technique for the solution of integral equations. It is widely used in the analysis of electromagnetic scattering problems [*Harrington*, 1968; *Peterson et al.*, 1998; *Rao et al.*, 1982; *Schaubert et al.*, 1984]. For a problem with *N* unknowns (number of basis functions for expansion of the unknown currents), the MoM results in a matrix equation with *N*^{2} matrix elements. The computational complexity for solving this matrix equation is *O*(*N*^{3}), with a direct matrix solver (LU-decomposition), or *O*(*N*^{2}) with an iterative matrix solver [*Peterson et al.*, 1998; *Sarkar and Arvas*, 1985]. The computational complexity of the iterative solution is based on the *O*(*N*^{2}) complexity to fill the matrix and for computation of the matrix-vector product. The fast multipole method (FMM) and its multilevel extension, the multilevel fast multipole algorithm (MLFMA), have been developed to reduce the memory and computation time of MoM, so that it may be applied to the analysis of scattering from electrically large targets [*Rokhlin*, 1983, 1990; *Coifman et al.*, 1993; *Brandt*, 1991; *Song and Chew*, 1994, 1995; *Song et al.*, 1997, 1998; *Chew et al.*, 2001]. It is important to note that the FMM and its multilevel implementation are not limited to high-frequency applications; in fact the FMM was originally developed for potential theory [*Rokhlin*, 1983]. Since the FMM is a special case of the MLFMA, in the remainder of the text we only mention the MLFMA, for which all results are presented.

[3] Integral equations in electromagnetic scattering problems are based on either a surface or volumetric formulation (or a combination of the two). The surface-integral formulation is applicable to perfectly conducting or homogeneous dielectric targets, and is formulated using the boundary conditions on the surface of the scatterers. The volume formulation is applicable to dielectric and/or permeable targets, and for a non-magnetic target the unknowns are equivalent volume electric currents defined in terms of the target permittivity, which may be heterogeneous. Volume integral equations use a volumetric mesh, and therefore the number of unknowns is proportional to *D*^{3}, where *D* is the principal dimension of the scatterer. As a result, the number of unknowns increases rapidly with the size of the scatterer and the volume integral equation technique is typically only applied to electrically small targets. In order to make volume integral equation techniques applicable to electrically large scatterers, the MLFMA has been exploited [*Lu et al.*, 2000]. However, this previous research has focused on free-space scattering problems. There are many applications for which the free-space models are not appropriate. For example, there is significant interest in radar-based remote sensing of buried targets such as land mines, unexploded ordnance (UXO), drums, tunnels, etc. The MLFMA has been extended previously to the half-space problem for the case of surface electric and combined-field integral equations [*Geng et al.*, 1999, 2000, 2001]. In this paper, we extend the MLFMA to volume integral equations, for scatterers in the vicinity of a half-space. The principal advantage of the volumetric formulation is found in its ability to handle general target heterogeneity.

[4] The main numerical challenge in addressing the half-space problem, vis-à-vis the free-space case, is found in evaluation of the half-space dyadic Green's function. The MLFMA clusters the basis and testing functions into a set of groups. Depending on the distance between these groups, the intergroup interactions are characterized as “near” or “far”, with this discussed in greater detail below. “Near” terms are handled in a manner analogous to a traditional MoM analysis, and here the “near” terms are calculated by rigorous evaluation of the dyadic half-space Green's function, using the complex-image technique (DCIT) [*Michalski and Zheng*, 1990; *Chow et al.*, 1991; *Yang et al.*, 1991; *Shubair and Chow*, 1993]. The “far” interactions are addressed approximately, but accurately, using image terms. The latter formulation is necessitated by the fact that the underlying relationship associated with the MLFMA is based on the free-space Green's function, augmented here via image theory.

[5] We present a detailed description of the implementation of the half-space MLFMA for a volumetric electric field integral equation (VEFIE) formulation. The volumetric formulation is compared to a half-space MLFMA analysis based on surface combined-field integral equations (SCFIE) of the first kind. The two approaches are compared using several numerical examples.