Seismic ray theory

Definition

Let an isotropic linearly elastic medium in a region of $\R^3$ have density $\rho(x) > 0$ and Lamé parameters $\lambda(x)$, $\mu(x)$, all smooth. Its P-speed and S-speed are $\alpha = \sqrt{(\lambda + 2\mu)/\rho}$ and $\beta = \sqrt{\mu/\rho}$. Seismic ray theory is the asymptotic approximation of time-harmonic solutions of the elastic wave equation, as the angular frequency $\omega \to \infty$, by a ray series $$u(x, t) = e^{i\omega (t - T(x))} \sum_{k \ge 0} \frac{A_k(x)}{(i\omega)^k}.$$ At leading order:

1. The travel time $T$ satisfies the eikonal equation $\abs{\nabla T}^2 = 1/v^2$, with $v = \alpha$ for a P-wave and $v = \beta$ for an S-wave.
2. The amplitude $A_0$ is parallel to $\nabla T$ for a P-wave and orthogonal to $\nabla T$ for an S-wave.
3. The rays are the curves $x(s)$, with $s$ the arc length, such that $dx/ds = v \nabla T$; they satisfy $$\frac{d}{ds}\Big(\frac{1}{v} \frac{dx}{ds}\Big) = \nabla \frac{1}{v}.$$