Derivatives of surface horizontal slownesses

Another set of variables involved in partial differential equation (4) are partial derivatives of surface horizontal slownesses. According to the definition in equation (2), we can calculate these dependent variables by using a finite difference approximation. In Cartesian coordinates,

$$\left\{ \begin{array} {lll} p_x(x,z;x_c) & \approx & \displa... ...e{{\Delta \tau_z(x,z;x_c) \over \Delta x_c}}\end{array}\right.$$

and in polar coordinates,

$$\left\{ \begin{array} {lll} p_r(r,\theta;x_c) & \approx & -\... ...laystyle{\Delta \tau_\theta \over \Delta x_c}\end{array}\right.$$

where x_c=x_s or x_r. There are two drawbacks to using these equations. The first is the property of the difference operators that I have mentioned before. The second disadvantage is that calculating the traveltime-changes with respect to the change of shot or receiver position requires us to compute the partial derivatives of traveltimes at least twice at the neighbor points, which significantly complicates the algorithm.

The solution to this problem is to find the partial differential equations for p. Appendix A shows that p satisfies the following partial differential equation:

$$\tau_xp_x+\tau_zp_z = 0,$$ (5)

which implies the orthogonal relation between rays and wavefront. The equivalent of this equation in polar coordinates is  

$$\tau_rp_r+{1 \over r^2}\tau_\theta p_\theta = 0.$$ (6)

To solve this equation, we can use the same algorithm as the one used for traveltime calculation. Let us use the equation in polar coordinates as an example. We define

$$u = p_\theta \ \ \ \hbox{and}\ \ \ w = p_r.$$

Equation (6) can then be rewriten as

$$w=-{\tau_\theta \over r^2 \tau_r}u.$$

Because

$${\partial^2 p \over \partial \theta \partial r} = {\partial^... ...rtial w \over \partial \theta} = {\partial u \over \partial r},$$

we have the transformed partial differential equation  

$${\partial u \over \partial r} = - {\partial \over \partial \theta} \left[{\tau_\theta \over r^2 \tau_r}u\right].$$ (7)

If s₀ is the slowness at the origin of polar coordinates, then the initial conditions are

$$p(0,\theta;x_c)=-s_0\sin\theta, \ \ \ \ \ \ \ \ u(0,\theta;x_c)=-s_0\cos\theta \ \ \ \ \hbox{and}\ \ \ \ w(0,\theta;x_c) = 0.$$

We can solve equation (7) by using the standard up-wind finite-difference method. The direct results are the partial derivatives of p which appear in equation (4).