The advection equation for take-off angle

The amplitude satisfies the zeroth order transport equation Cervený et al. (1977):

$$\begin{eqnarray} \nabla \tau \cdot \nabla A + \frac{1}{2} A \nabla^{2}\tau &=& 0.\end{eqnarray}$$ (3)

If the traveltime field has been found by solving the eikonal equation, then equation (3) is a first order advection equation. However, we see that the Laplacian of traveltime field is involved in the transport equation, which implies that we need a third order accurate traveltime field to get a first order accurate amplitude field El-Mageed et al. (1997); El-Mageed (1996); Symes (1995). To avoid this complexity, we use another approach to compute the amplitude.

In 2D isotropic media, the amplitude satisfies Cervený et al. (1977); Friedlander (1958)

$$\begin{eqnarray} A&=& \frac{v}{2\pi \sqrt{2}\sqrt{\vert J\vert}} \nonumber \\ &... ...{v}{2\pi \sqrt{2}}\sqrt{\nabla \tau \times \nabla \phi}, \nonumber\end{eqnarray}$$

where J(x,z;x_s,zs) is the Jacobian of the transformation from Cartesian coordinates (x,z) to ray coordinates $(\tau,\phi)$, $\tau$ is the traveltime, $\phi=\phi (x,z;x_{s},z_{s})$ is the take-off angle from source point (x_s,z_s) to a general point (x,z) in the subsurface:

$$\begin{eqnarray} J&=& \left \vert \begin{array} {cc} \frac{\partial x}{\partial... ...mber \\ &=& \frac{1}{\nabla \tau \times \nabla \phi}, \nonumber\end{eqnarray}$$

where $\nabla \phi$ and $\nabla \tau$ are the gradients of take-off angle and traveltime, respectively.

Since the take-off angle $\phi$ is constant along any ray,

$$\begin{eqnarray} \nabla \tau \cdot \nabla \phi = \frac{\partial \tau}{\partial x... ...{\partial \tau}{\partial z}\frac{\partial \phi}{\partial z} &=& 0.\end{eqnarray}$$ (4)

That is, the gradient $(\frac{\partial \tau}{\partial x},\frac{\partial \tau}{\partial z})$ is the wavefront normal which is tangential to the ray; the gradient $(\frac{\partial \phi}{\partial x},\frac{\partial \phi}{\partial z})$ is tangential to the wavefront.

However, the gradient of the take-off angle depends on the second order derivative of traveltime, so that we need third order accurate traveltimes to get a first order accurate gradient of take-off angle. Zhang 1993 used this equation in polar coordinates to compute the geometrical spreading factor, but his computation of the take-off angle was based on the first order traveltime field. Consequently, the gradient of take-off angle computed by his scheme was inaccurate. Vidale 1990 encountered a similar difficulty.