The WKBJ Green's tensor

The Green's tensor ${\bf {\cal G}}({\bf x},t;{\bf x}',t')$ is the displacement vector impulse response at an ``observation'' point $({\bf x},t)$ due to an impulsive force density located at $({\bf x}',t')$, and satisfies the auxiliary wave equation:

 

$${\bf {\cal L}}{\bf {\cal G}}({\bf x},t;{\bf x}',t') = {\bf {\cal I}}\d({\bf x}-{\bf x}') \d(t-t')\;,$$ (13)

where ${\bf {\cal I}}$ is the identity matrix. I will assume the following WKBJ form for the Green's tensor:

$$\begin{eqnarray} \lefteqn{{\bf {\cal G}}({\bf x},t;{\bf x}',t') \sim } \nonumber... ...{\bf x}') + {\bf e}_3^n({\bf x}){\bf e}_3^n({\bf x}')\right] \;,\end{eqnarray}$$ (14)

where ${\bf e}_k^n$ are the polarization vectors evaluated at either endpoint along the ray joining ${\bf x}'$ to ${\bf x}$, and where $\{A,B,\tau,\phi\}$ all have the functional dependency $A({\bf x};{\bf x}')$. I refer the reader to Aki and Richards (1980) for a constant velocity overview of the Green's tensor, and to Morse and Feshbach (1953) for a review of the WKBJ method of solution to partial differential equations. Note that the form (14) of ${\bf {\cal G}}$ implicitly requires an isotropic elastic background propagation model since the two shear components have the same arrival time $\phi$ and amplitude B, although all of the theory previous to this point has been valid for general anisotropy. The traveltimes $\tau$ and $\phi$ satisfy the P and S eikonal equations respectively,

$$\vert \nabla\tau({\bf x}) \vert^2 = \alpha^{-2}({\bf x})$$

 

$$\vert \nabla\phi({\bf x}) \vert^2 = \beta^{-2}({\bf x}) \;,$$ (15)

and the WKBJ amplitudes A and B satisfy the P and S transport equations respectively,

$$A({\bf x})\nabla^2\tau({\bf x}) + 2\nabla\tau({\bf x})\cdot\nabla A({\bf x}) = 0$$

 

$$B({\bf x})\nabla^2\phi({\bf x}) + 2\nabla\phi({\bf x})\cdot\nabla B({\bf x}) = 0 \;.$$ (16)

The P and S-wave propagation velocities are denoted as $\alpha({\bf x})$ and $\beta({\bf x})$ respectively. Incorporating general anisotropy in the background model would require modification of the Green's tensor (14) to have two coupled shear traveltimes $\phi_1, \phi_2$ and amplitudes B₁, B₂, given by S₁-S₂ coupled eikonal and transport equations (Cervený et al., 1977). Since the WKBJ Green's tensor satisfies the zeroth order asymptotic ray eikonal and transport equations, the quantities A, B, $\tau$, and $\phi$ can be evaluated by numerical ray techniques, providing the migration background model is ray valid (smooth velocity gradients on the order of a seismic wavelength within a sequence of possibly discontinuous parameterized layers).