APPENDIX A

In this appendix, I derive equations (13) and (15), and the initial conditions associated with them.

The Cartesian coordinates of the trajectory of a ray are functions of so-called ray coordinates (s,r), as follows:

$$\left\{ \begin{array} {lll} x & = & \hat{x}_i(s,r) \\ z & = & \hat{z}_i(s,r),\end{array}\right. \eqno(A.1)$$

where s measures the arc-length of the ray and r is the depth at which the ray reaches the receiver well. Subscript i indicates the source number. If we solve equation (A.1) for r, we obtain a function $r=\hat{r}_i(x,z)$ that has a constant value along each ray. Consequently, the gradient directions of function $\hat{r}_i(x,z)$ are orthogonal to rays. Because the gradient directions of traveltime function $\tau_i(x,z)$ are tangential to rays, we have the relation $\nabla\tau_i\cdot \nabla\hat{r}_i=0$,which, in two dimensions, yields

$${\partial \tau_i \over \partial x}{\partial \hat{r}_i \over ... ... \partial z}{\partial \hat{r}_i \over \partial z}=0. \eqno(A.2)$$

At any receiver location, function $\hat{r}_i(x,z)$ is equal to the depth of the receiver, which acts as an initial condition required in solving equation (A.2).

To derive equation (15), we first define

$$q_i(x,z) = \int^{s_i(x,z)}_0 \sum^M_{j=1}g_j\beta_j(\hat{x}_i(\xi,r), \hat{z}_i(\xi,r))d\xi. \eqno(A.3)$$

At the source location, s_i(x,z) is equal to zero; hence q_i(x,z) is equal to zero. At the receiver locations, s_i(x,z)=s_i(r). Comparing equation (A.3) with equation (15), we can conclude that q_i(x,z) is equal to h⁽¹⁾(r) at the receiver locations. Now, what remains is to show that q_i(x,z) defined in equation (A.3) satisfies the first-order linear partial differential equation (15). To do so, we take the derivative of both sides of equation (A.3) with respect to s, which yields

$${\partial q_i \over \partial x}{dx \over ds}+ {\partial q_i ... ...m^M_{j=1}g_j\beta_j(\hat{x}_i(s,r),\hat{z}_i(s,r)). \eqno(A.4)$$

Functions $x=\hat{x}_i(s,r)$ and $z=\hat{z}_i(s,r)$ define the Cartesian coordinates of the ray. They satisfy the ray equation as follows:

$$\left\{ \begin{array} {lll} \displaystyle{dx \over ds} & = &... ...partial z} \right] & = & m_z(x,z),\end{array}\right. \eqno(A.5)$$

where subscripts x and z denote partial derivatives with respect to x and z, respectively. Substituting these functions into equation (A.4) gives

$${\partial \tau_i \over \partial x}{\partial q_i \over \parti... ...over \partial z}= m(x,z)\sum^M_{j=1}g_j\beta_j(x,z), \eqno(A.6)$$

which is equation (15).