Partial differential equations

To use a finite-difference approach, we need to find the expressions for partial derivatives:

$${\partial \hat{x} \over \partial x} \ \ \ \ \hbox{and}\ \ \ \ {\partial \hat{z} \over \partial x}$$

or

$${\partial \hat{x} \over \partial z} \ \ \ \ \hbox{and}\ \ \ \ {\partial \hat{z} \over \partial z}.$$

In my preceding paper (Zhang,1990b), I showed that  

$$\left\{ \begin{array} {lll} \displaystyle{{\partial \hat{x} ... ...\hat{z}}-\hat{F}_{\hat{z}}\hat{G}_{\hat{x}}}}\end{array}\right.$$ (3)

and  

$$\left\{ \begin{array} {lll} \displaystyle{{\partial \hat{x} ... ...\hat{z}}-\hat{F}_{\hat{z}}\hat{G}_{\hat{x}}}}\end{array}\right.$$ (4)

where

$$\begin{array} {lll} \hat{F}_{\hat{x}} & = & \hat{\tau}_{\hat... ...yle{\partial x_s \over \partial x_r}+ p_z(x,z;x_r).\end{array}$$

If the traveltime table is calculated with polar coordinates $(r,\theta)$,instead of Cartesian coordinates (x,z), then we can simply change x and z in equations (3) and (4) to $\theta$ and r, respectively.

With equations (3) and (4), we can extrapolate the operators in both the horizontal and vertical directions. Because the operators are usually known at the surface, one can use equation (4) to extrapolate the operators vertically to each depth. The next section shows how this extrapolation is done.