Computation of matrix ${\bf A}$

The elements of matrix ${\bf A}$ are the derivatives of the traveltimes with respect to the slowness model parameters. Let us define

$$p_{ij}(x,z)={\partial \tau_i \over \partial m_j}.$$ (8)

Taking the derivatives of both sides of the eikonal equation with respect to m_j gives  

$${\partial \tau_i \over \partial x}{\partial p_j \over \parti... ...r \partial z}{\partial p_j \over \partial z} =m(x,z)\beta(x,z).$$ (9)

If we assume that $\tau_i(x,z)$ has been computed by solving the eikonal equation, then equation (9) is a first-order linear partial differential equation (PDE). Because the traveltime at the source location is always zero no matter how the slowness model changes, equation (9) should be solved with the initial condition that p_ij(x,z)=0 at the source location. Evaluating the function p_ij(x,z) at receiver locations gives the elements of matrix ${\bf A}$.