## 1. Introduction

[2] Electromagnetic (EM) propagation at low frequencies (EM diffusion) is used in a number of applications, such as geothermal exploration [*Pellerin et al.*, 1996], evaluation of hydrocarbon resources by mapping the subseafloor resistivity [*Eidesmo et al.*, 2002], EM induction in boreholes and logging while drilling [*Wang and Signorelli*, 2004], magnetotelluric problems [*Mackie et al.*, 1993; *Yin and Maurer*, 2001], and geoelectrical surveys for groundwater and mineral exploration [*Oristaglio and Hohmann*, 1984].

[3] The theory of EM diffusion in isotropic media is well established [see, e.g., *Ward and Hohmann*, 1988]. Anisotropy has been taken into account to model magnetotelluric fields, using a propagation matrix algorithm in 1-D layered models, where the conductivity is homogeneous both laterally and vertically within each layer [*Mann*, 1965; *Loewenthal and Landisman*, 1973; *Abramovici*, 1974; *Kováčiková and Pek*, 2002]. In all these works no analysis of the physics in 3-D space is performed. At most, the wave numbers along the vertical direction are obtained. This is because they only consider the *z* direction and then there is no dependence with the propagation angle in their equations. In fact, the study of anisotropic diffusion in three dimensions and from the point of view of the energy balance has not given much attention in the geophysical literature. An analysis has been performed by *Carcione and Schoenberg* [2000] and *Carcione* [2007], who considered the high-frequency range (waves), where the dielectric permittivity plays an important role. Analogies can be performed with the theory of elasticity to establish mathematical and physical formulations [*Carcione and Cavallini*, 1995; *Carcione and Helbig*, 2008].

[4] There are many material configurations in the subsurface that might lead to anisotropy [*Negi and Saraf*, 1989]. The geophysical motivation behind the use of an electrically anisotropic description of the Earth are given by *Mann* [1965], *Carcione* [1996], *Weidelt* [1998], *Anderson et al.* [2001], *Carcione and Schoenberg* [2000], *Wang and Fang* [2001], and *Weiss and Newman* [2002]. It might be that there are some preferred directions in the subsurface rocks, or some preferred orientation of grains in the sediments. Compaction, fine layering or a pronounced strike direction might lead to effective anisotropy. Alternations of sandstone and shales may give reservoir anisotropy [*Carcione and Seriani*, 2000; *Davydycheva et al.*, 2003].

[5] The paper is organized as follows. First, Maxwell's equations are given, in the inhomogeneous and homogeneous cases. Then, I perform a plane-wave analysis and obtain the Kelvin-Christoffel eigensystem, whose eigenvalues yield the phase velocities and skin depth as a function of the conductivity components, frequency and propagation direction. The energy balance (Umov-Poynting theorem) is then established to obtain expressions of the energy densities and energy velocity. Finally, I obtain the Green function for a uniaxial medium.