Traveltimes

The eikonal equation in two dimensions relates the two gradient components of the traveltime field as follows:

$$\tau^2_x(x,z)+\tau^2_z(x,z)=m^2(x,z),$$ (3)

where subscripts x and z denote partial derivatives with respect to x and z, respectively, and m(x,z) is the slowness model of the medium determined by

$$m(x,z)= \left\{ \begin{array} {ll} \sqrt{\displaystyle{\rho(... ... \over \mu(x,z)}} & \ \ \hbox{for $S$-waves}.\end{array}\right.$$ (4)

Traditionally, the eikonal equation is solved by using the method of characteristics. The characteristic curves of this equation are rays described by the ray equations  

$$\left\{ \begin{array} {lll} \displaystyle{dx \over ds} & = &... ...\ \\ \displaystyle{d\tau_z \over ds} & = & m_z(x,z),\end{array}$$ (5)

where variable s is the arc-length of the ray. The solutions to the ray equation define a transformation between the coordinate system (x,z) and a new coordinate system $(s,\gamma)$, named ray coordinates, as follows:  

$$\left\{ \begin{array} {lll} x & = & \hat{x}(s,\gamma) \\ \\ z & = & \hat{z}(s,\gamma)\end{array}\right.$$ (6)

For each fixed $\gamma$, this pair of equations give the Cartesian coordinates of a ray. The variations of $\gamma$ span a set of rays covering the whole (x,z) space. If we substitute (x,z) in equation (6) into the eikonal equation, we obtain an ordinary differential equation (ODE) along each ray:

$${d\tau \over ds} = m(\hat{x}(s,\gamma),\hat{z}(s,\gamma)).$$ (7)

The solution of this ODE is simply an integral along the ray:

$$\tau(\hat{x}(s,\gamma),\hat{z}(s,\gamma))= \tau(\hat{x}(s_0,... ...))+ \int^s_{s_0}m(\hat{x}(\xi,\gamma),\hat{z}(\xi,\gamma))d\xi.$$ (8)

This traveltime calculation method is called ray tracing method. The ray tracing method computes traveltimes along rays, hence generates a traveltime map on an irregular grid.

Many seismic modeling and imaging algorithms require a traveltime map on a regular grid. An efficient way to obtain this traveltime map is to solve the eikonal equation directly using a finite difference scheme. The finite difference calculation of traveltimes is actually an extrapolation process. The traveltime function is locally extrapolated from the grid points where the traveltimes are known to their neighboring grid points where the traveltimes are unknown by following the principle that the gradients of the traveltimes satisfy the eikonal equation. This process starts from the source position where the traveltime is zero, and is repeated until all grid points are filled. We can also extrapolate the gradients of traveltimes. The gradients of traveltimes are vectors and have two components in two dimensions. In order to determine both components, we need one more equation besides the eikonal equation. Van Trier and Symes (1990) use the following one:  

$${\partial \tau_z \over \partial x} = {\partial \tau_x \over \partial z},$$ (9)

where $(\tau_x,\tau_z)$ are the two components of the traveltime gradient. Van Trier and Symes interpret this equation using the flux conservation principle. If we rewrite equation (9) as $\nabla \times \nabla \tau=0$, we see that this equation states that a gradient field is curl-free. Once the traveltime gradients are computed, their integrations give the traveltimes.