I present a plane-wave analysis of anisotropic electromagnetic media at the low-frequency range, where the displacement currents can be neglected and the field is diffusive. Anisotropy is due to the conductivity tensor and the magnetic permeability is a scalar quantity. The analysis includes the energy balance (Umov-Poynting theorem) and provides expressions of measurable quantities such as the phase and energy velocities, the attenuation factor, and the skin depth as a function of frequency and propagation direction. The balance of energy allows the identification of the stored and dissipated energy densities, which are related to the magnetic energy and the conductive part of the electric energy. For a real conductivity tensor, the stored energy equals the dissipated energy. I also establish fundamental relations, e.g., the scalar product between the slowness vector and the power-flow vector is equal to the energy density. For uniform plane waves, the phase velocity is the projection of the energy velocity vector onto the propagation direction and a similar relation is obtained by replacing the energy velocity with a velocity related to the dissipated energy. I have also obtained the Green function for an azimuthally isotropic medium (transverse isotropy), which is used to calculate transient fields.
 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  and Carcione , 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].
 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.
2. Maxwell's Equations for General Anisotropic Media
 In vector notation, Maxwell's equations, neglecting the displacement currents, are [e.g., Carcione, 2007]
where the vectors E, H, B, JI, JS, and M are the electric field intensity, the magnetic field intensity, the magnetic flux density, the induced current density, the electric source current, and the magnetic source density, respectively. In general, they depend on the Cartesian coordinates (x1, x2, x3) = (x, y, z), and the time variable t. I have used the compact notation ∂t ≡ ∂/∂t.
 Additional constitutive equations are needed. These are JI = σ · E and B = μH, where σ, the conductivity tensor, is a real, symmetric and positive definite tensor, and μ, the magnetic permeability, is a scalar quantity, and the dot denotes ordinary matrix multiplication. Substituting the constitutive equations into equations (1) gives
which are a system of six scalar equations in six scalar unknowns.
 In this work, I am concerned with triclinic media, where at each point of the space the conductivity tensor is nondiagonal. However, the tensor can always be rotated to obtain its expression in its principal coordinate system
Although the magnetic permeability is commonly assumed to be that of free space, some soils have a significantly higher value [Olhoeft and Capron, 1994], hence, μ can vary arbitrarily in space. In some cases, such as the CSEM problems μ is assumed to be spatially constant [Eidesmo et al., 2002].
 From equation (1) I obtain a vector equation for the electric field
which is valid for ∇ ·JS = 0, according to equation (1), i.e., if the source current density is divergence free [Ward and Hohmann, 1988, equation 1.7]. This condition is satisfied by inductively coupled magnetic dipole controlled sources of the form
where A is a vector. In 2-D (x, z) space, an example is Js = (∂3F, −∂1F)⊺h, where F(x) is the spatial distribution and h(t) is the time history.
 In homogeneous media with a diagonal conductivity tensor, equation (4) becomes
 Consider the 2D anisotropic case, ignoring the y dimension. The corresponding equations are
From (6) one has σ1∂1E1 + σ3∂3E3 = 0. Combining this equation with (10) yields
is a modified Laplacian. Note that the field components have decoupled.
 In 3D space, the E3 component can be decoupled if one considers a transversely isotropic medium with σ1 = σ2. In this case, a similar procedure leads to
and a corresponding analytical solution is given below.
3. Plane-Wave Theory
 I assume nonuniform harmonic plane waves with a phase factor
where ξ, the complex slowness vector, is equivalent to k/ω, with k and ω being the wave number vector and frequency, respectively. The dot denotes the scalar product and = . Note the following correspondences between time and frequency domain:
where × denotes the vector product.
 Substituting the plane wave (15) into Maxwell's equations (2) in the absence of sources, and using (16) gives
For convenience, the medium properties are denoted by the same symbols, in both the time and frequency domains.
for three equations for the components of E. Alternatively, the vector product of equation (18) with ξ and use of (17) yields
for three equations for the components of H.
 The equivalent of the 3 × 3 viscoelastic Kelvin-Christoffel equations, for the electric field vector components, are
where the subindices take the values 1, 2 and 3, and eijk are the elements of the Levi-Civita tensor.
 Similarly, the equations for the magnetic field vector components are
Rotating the conductivity tensor to the principal system of coordinates, equation (3) is obtained. There is no loss of generality in this operation. Then, the equivalent of the viscoelastic Kelvin-Christoffel equation for the electric field vector is
where the EM Kelvin-Christoffel matrix is
the dispersion relation (i.e., the vanishing of the determinant of the Kelvin-Christoffel matrix) becomes
There are only quartic and quadratic terms of the slowness components in the dispersion relation.
3.1. Slowness, Kinematic Velocities, Attenuation, and Skin Depth
 The slowness vector can be split into real and imaginary vectors such that ωRe(t − ξ · x) is the phase and ωIm(ξ·x) is the attenuation. Assume that propagation and attenuation directions coincide to produce a uniform plane wave, which is equivalent to a homogeneous plane wave in viscoelasticity. The slowness vector can be expressed as
where ξ is the complex slowness and = (l1, l2, l3)⊺ is a real unit vector, with li the direction cosines (li2 = 1). I obtain the real wave number vector and the real attenuation vector as
 The skin depth is the distance d for which exp(−α·d) = 1/e, where e is Napier's number, i.e., the effective distance of penetration of the signal, where α = α. Using equation (29) yields
since di = dli.
 Assume, for instance, propagation in the (1, 2) plane. Then, l3 = 0 and the dispersion relation (30) is factorizable, giving
These factors give the TM and TE modes with complex velocities
In the TM (TE) case the magnetic (electric) field vector is perpendicular to the propagation plane. For obtaining the slowness and complex velocities for the other planes, simply make the following subindex substitutions:
The analysis of all three planes of symmetry gives the curves represented in Figure 1b. There exists a single conical point given by the intersection of the TE and TM modes, as can be seen in the (1,2) plane of symmetry. The location of the conical point depends on the values of the material properties. The slowness curves of the TE modes intersecting the three orthogonal planes are circles (isotropy).
 Consider the TE mode. Then, the phase velocity is
where f = ω/(2π) is the frequency. On the other hand, the skin depth and attenuation factor are given by
The velocity increases and the skin depth decreases with frequency. In the anisotropic case, these quantities depend on the propagation angle.
3.2. Umov-Poynting's Theorem and Energy Velocity
 The scalar product of the complex conjugate of equation (18) with E, use of the relation 2Im(ξ) · (E × H*) = (ξ × E) · H* + E · (ξ × H)* and substitution of equation (17), gives Umov-Poynting's theorem for plane waves
is the complex Umov-Poynting vector, and
are the electric (dissipated) and magnetic (stored) energy densities, respectively. The superscript “*” denotes complex conjugate. The imaginary part of equation (41) gives the balance of stored energy and the real part gives the balance of dissipated energy. This equation hold for a complex conductivity tensor, i.e., including induced polarization effects [Zhdanov, 2008]. In this work, σ is real and um and ue are therefore real quantities. Because of this fact, there is no electric stored energy in the diffusion process. Furthermore, it is shown in the appendix that
is a velocity associated with the dissipated energy. Similar relations are obtained in viscoelasticity and poroviscoelasticity [e.g., Carcione, 2007]. The first equation (51) indicates that the slowness and energy velocity surfaces are reciprocal.
3.4. The TE and TM Modes
 As an example, I consider the TE mode propagating in the (1,2) plane. Then,
 The solution corresponds to transverse isotropy, for which there are two eigen directions, and only two, for which all three tensors have equal eigenvalues. This electromagnetic symmetry includes that of hexagonal, tetragonal and trigonal crystals. These are said to be optically uniaxial.
 I consider σ1 = 0.2 S/m, σ2 = 0.03 S/m and σ3 = 0.05 S/m, and μ = μ0 = 4 π 10−7 H/m. The phase velocity, energy velocity and skin depth at the Cartesian planes are shown in Figure 1. The frequency is f = 1 Hz. The electric (magnetic) field is perpendicular to the plane for the TE (TM) mode (see Figure 1b). The first are isotropic while the TM modes show anisotropy. Both the velocity and skin depth decrease for increasing conductivity. This means, for instance, that the diffusion process is slower in salt water than in fresh water, or slower in brine-saturated sediments than in oil-saturated reservoirs. On the other hand, the velocity increases and the skin depth decreases with the frequency.
 Let us consider a transient source, whose time history is
where tp is the period of the wave (the distance between the side peaks is tp/π) and I take ts = 1.4tp. I consider σ1 = σ2 = 0.1 S/m and σ3 = 0.05 S/m and a central frequency fp = 3 Hz. Figure 2 shows the time history at two different distances from the source location, corresponding to the anisotropic and isotropic cases (solid and dashed lines, respectively). The electric field is normalized with respect to maximum amplitude. Isotropy corresponds to σ1 = σ3 = 0.1 S/m. The amplitude and phase variations with offset are shown in Figure 3. As can be appreciated, the differences can be substantial.
 The study of electromagnetic propagation in anisotropic media requires a detailed plane-wave analysis and the establishment of the energy balance to obtain the expression of measurable quantities such as the energy velocity as a function of frequency and propagation direction. In the case of uniform plane waves, the stored and dissipated energies have the same value. Fundamental relations are obtained, e.g., the energy density can be obtained as the scalar product of the slowness and the power-flow vectors. For uniform waves, one of the relations indicates that the energy velocity and slowness surfaces are reciprocal.
 The theory can be generalized to the case of induced polarization, i.e., a complex and frequency-dependent conductivity tensor. The generalization is straightforward. In this case, the stored electric energy is not zero and the magnetic stored energy does not have the same value of the dissipated energy. In terms of mechanical models, it can be shown that the conductivity components can be represented by Kelvin-Voigt elements.
 A closed form solution is obtained in the time domain for a uniaxial (transversely isotropic) medium. The solution is useful to obtain amplitude and phase variations as a function of the propagation distance (offset) in the frequency domain.
 The examples illustrate the differences between the anisotropic and isotropic cases, which can be significant.
Appendix A:: Equivalence Between the Stored and Dissipated Energy Densities
 I show here that, for uniform plane waves and a real conductivity tensor, the stored energy equals the dissipated energy over a cycle. The magnetic energy (43) is
where I have used equation (17) and the property (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C). On the other hand, the electric energy (46) is
where I have used equation (20), the fact that σ is real and the property A × (B × C) = (A · C)B − (A · B)C.
 From equations (31) and (32), it is ξ = (1 − )s and ∣ξ∣2 = ξ · ξ. Then, the last equation and ∣ξ · E*∣2 = (ξ·E)(ξ · E*) imply