an equivalent medium theory, a heterogeneous medium can be replaced by a homogeneous one that predicts wave propagation in the actual medium. Backus showed that layering on a scale much finer than the wavelength has an impact and that a number of isotropic layers can be replaced by a homogeneous transversely isotropic medium that behaves exactly in the same manner as the actual medium under static load in the infinite wavelength limit. If each layer is described by 5 transversely isotropic parameters , specifying the matrix The elastic moduli for the effective medium will be where denotes the volume weighted average over all layers. This includes isotropic layers, as the layer is isotropic if , and . Short and medium wavelength approximation Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave. However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle are. The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given by where is the angle between the axis of symmetry and the wave propagation direction, is mass density and the are elements of the elastic stiffness matrix. The Thomsen parameters are used to simplify these expressions and make them easier to understand. Thomsen parameters Thomsen parameters are dimensionless combinations of elastic moduli that characterize transversely isotropic materials, which are encountered, for example, in geophysics. In terms of the components of the elastic stiffness matrix, these parameters are defined as: where index 3 indicates the axis of symmetry () . These parameters, in conjunction with the associated P wave and S wave velocities, can be used to characterize wave propagation through weakly anisotropic, layered media. Empirically, the Thomsen parameters for most layered rock formations are much lower than 1. The name refers to Leon Thomsen, professor of geophysics at the University of Houston, who proposed these parameters in his 1986 paper "Weak Elastic Anisotropy". Simplified expressions for wave velocities In geophysics the anisotropy in elastic properties is usually weak, in which case . When the exact expressions for the wave velocities above are linearized in these small quantities, they simplify to where are the P and S wave velocities in the