Computation of the conjugate gradient vector

The conjugate gradient vector has two parts. The first part ${\bf h}^{(1)}(r)$is related to the traveltime residuals, and the second part ${\bf h}^{(2)}(r)$ is related to the constraints. The computation of ${\bf h}^{(2)}(r)$ is simple because matrix ${\bf B}$ is known. We want to compute ${\bf h}^{(1)}(r)$ with a finite difference method. The components of ${\bf h}^{(1)}(r)$ are as follows:

$$\begin{array} {lll} h^{(1)}_i(r) & = & \displaystyle{\sum^M_... ...j \beta_j(\hat{x}_i(\xi,r), \hat{z}_i(\xi,r))d\xi}.\end{array}$$ (14)

It is shown in Appendix A that if we solve the first-order linear PDE  

$${\partial \tau_i \over \partial x}{\partial q_i \over \parti... ...artial q_i \over \partial z}= m(x,z)\sum^M_{j=1}g_j\beta_j(x,z)$$ (15)

with the initial condition that q_i(x,z) is equal to zero at the source location, then h⁽¹⁾_i(r) is equal to q_i(x,z) at the receiver locations. Equation (15) has a form similar to the eikonal equation; hence we can use the algorithm for the traveltime calculation, after being slightly modified, to solve it.