In this appendix, I derive the formula for calculating the Jacobian function J(x,z) and the partial differential equation (14).
From equation (12), we have
The expression in the denominator of the last line is a cross-product of two gradient vectors:
is the take-off angle of the ray that reaches point(x,z), and is constant along each ray. Thus, the gradient directions of are always orthogonal to rays. On the other hand, the gradient directions of traveltimes are always tangential to rays. Consequently, and are orthogonal, which yields
In 2-D, equation (A.3) can be rewritten as
which is equation (14). From equations (A.1), (A.2) and (A.4), we can derive
The eikonal equation states that or . Applying this relation to equation (A.6), we get equation (12):