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:
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 that has a constant value along each ray. Consequently, the gradient directions of function are orthogonal to rays. Because the gradient directions of traveltime function are tangential to rays, we have the relation ,which, in two dimensions, yields
At any receiver location, function 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
At the source location, si(x,z) is equal to zero; hence qi(x,z) is equal to zero. At the receiver locations, si(x,z)=si(r). Comparing equation (A.3) with equation (15), we can conclude that qi(x,z) is equal to h(1)(r) at the receiver locations. Now, what remains is to show that qi(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
Functions and define the Cartesian coordinates of the ray. They satisfy the ray equation as follows:
where subscripts x and z denote partial derivatives with respect to x and z, respectively. Substituting these functions into equation (A.4) gives
which is equation (15).