Following Biondi's 2005 demostration, the derivative of the image depth () with respect to the subsurface-offset (), at a constant image midpoint (), and the derivative of the depth with respect to the image point, at a constant subsurface offset are given by the following:
where the partial derivatives are:
Figure presents the analytical solutions for the tangent to the impulse response. This was done for an impulse at a PS-travel time of 2 s, and a value of 2. The left panel shows the solution for equation . The right panel shows the solution for equation . The solid lines superimpose on both surfaces represents one section of the numerical derivative to the impulse response. The perfect correlation between the analytical and numerical solution validates our analytical formulations. This results supports the analysis presented with the kinematic equations (Appendix A).