Ray-coordinate wavefield extrapolation

Wavefield extrapolation in ray-coordinates requires casting the acoustic wave-equation not in the usual Cartesian representation, but rather in a system parameterized by phase-ray variables. In 2-D, these variables consist of $\t$, the one-way travel time from a source/receiver point along the direction of a ray, and $\gamma$,the direction across the rayfront at a constant time step. A cartoon illustrating ray-coordinate geometry is presented in Figure 1.

 

raycoord Figure 1 Cartoon illustrating the phase-ray coordinate system and its relation to the Cartesian basis. Variable $\t=\t(x,z)$ is the direction along a single ray, and parameter $\t^t$ is an isochron or rayfront. Variable $\gamma=\gamma(x,z)$ is the coordinate across the rayfront at a constant $\t$ step, and parameter $\gamma^g$ is a ray. Grey lines illustrate the mapping between ray point $(\ti1,\gi1)$ and Cartesian point (xo,zo). Angle $\theta$ is a rotation angle between the ray and the z-axis (assumed to be positive downward).

Note that the dimensions of time and space coordinates $\t$ and $\gamma$are seconds and meters, respectively.

The 2-D acoustic wave-equation for wavefield, ${\cal U}$, at frequency, $\omega$, in ray-coordinates is Sava and Fomel (2003),  

$$\frac{1}{v J} \left[ \frac{\partial}{\partial\t} \left( \fra... ...partial\gamma} \right) \right]= -\frac{\omega^2}{v^2} {\cal U},$$ (6)

where v is the velocity function, and J is the ray-coordinate Jacobian or geometrical ray spreading factor given by,  

$$J=\left[ \frac{\partial x_k}{\partial\gamma} \frac{\partial x_k}{\partial\gamma} \right]^ {\frac{1}{2}}.$$ (7)

Importantly, parameter J is solely a component of ray-coordinates and is independent of wavefield extrapolated on the coordinate system.

Analogous to wave-equation extrapolation in Cartesian coordinates, a dispersion relation must be specified that forms the basis for all derived ray-coordinate extrapolation operators. The relation being sought is the wavenumber along the ray direction, $k_\tau$.Following Sava and Fomel (2003), the partial derivative operators in equation (6) are expanded out to generate a second-order partial differential equation with non-zero cross derivatives. Fourier-domain wavenumbers are then substituted for the partial differential operators acting on wavefield, ${\cal U}$, and the quadratic formula is applied to yield the expression for $k_\tau$,  

$$k_\tau= \frac{{\rm i}v}{2 J} \frac{\partial}{\partial \tau}\... ...t) k_\gamma- \frac{v^2}{J^2} k_\gamma^2 \right]^{\frac{1}{2}}.$$ (8)

Note, again, that quantity $\frac{J}{v}$ depends solely on the coordinate system and is independent of the wavefield being propagated.

One relatively straightforward manner to apply wavenumber $k_\tau$in an extrapolation scheme is to develop the ray-coordinate equivalent of Claerbout's classic 15 equation Claerbout (1985). This involves a second-order Taylor series expansion of the radical in equation (8), and the identification of Fourier dual parameters $k_\tau$ and $k_\gamma$ with their space domain derivative counterparts $-{\rm i}\frac{\partial}{\partial \tau}$ and $-{\rm i}\frac{\partial}{\partial \gamma}$. The ray-coordinate formula corresponding to the 15equation is,

$$\begin{eqnarray} \frac{\partial{\cal U}}{\partial\t} \approx &-& \frac{v}{2J}\fr... ...c{v^2}{J^2} \right] \frac{\partial^2 {\cal U}}{\partial \gamma^2},\end{eqnarray}$$ (9)

where $\omega_o$ may be considered as the effective (non-stationary) frequency,

$$\omega_o = \omega\left[ 1 - \left( \frac{v}{2 \omega J} \fra... ...tau}\left( \frac{J}{v}\right) \right)^2 \right]^{\frac{1}{2}}.$$ (10)

Equation (9) may be solved in 2-D using fully implicit finite difference methods (e.g. Crank-Nicolson) and fast tridiagonal solvers. After wavefield solution, ${\cal U}(\t,\gamma,\omega)$, has been computed at all rayfield locations, the result is mapped to Cartesian coordinates using sinc-based interpolation operators in a neighborhood about each mapped point.