THEORY

The elastic wave equation for an isotropic, 2-dimensional medium has a well-known form:  

$$\begin{array} {lll} \rho \displaystyle{\partial^2 u_1 \over ... ...displaystyle{\partial u_1 \over \partial z})\right],\end{array}$$ (1)

where u₁ and u₂ are the x- and z-components of the displacement vector, $\lambda=\lambda(x,z)$ and $\mu=\mu(x,z)$ are the elastic constants of the media, and $\rho$ is the density of the medium. Using asymptotic ray theory, we can expand the Green's function of this wave equation into a ray series at high frequencies (Cervený et al., 1977). The leading term of this ray series has two components: G_P for the compressional wave (P-wave) and G_S for the shear wave (S-wave):  

$$G_c(\omega,x,z;x_0,z_0)=A_c(x,z;x_0,z_0)e^{i\omega\tau_c(x,z;x_0,z_0)}. \ \ \ \ c=P\ \hbox{or}\ S$$ (2)

In this equation, $\tau_c$ is the traveltime it takes for a wave to propagate from a source location (x₀,z₀) to an observation location (x,z). The traveltime function satisfies the eikonal equation of wave equation (1). If we assume that the medium is smoothly varying, then function A_c describes the amplitude decay of this propagation due to geometric spreading, and it satisfies the first order transport equation. Because the Green's functions for P-wave and S-wave have a common form, from now on, we drop the subscripts P and S, and make distinctions only when necessary.