Computation of Traveltime for refracted rays

In this Appendix I derive equations 5 and 6. From equation 3 we have:  

$$t_{s_1}\cos\alpha_s+\rho\tilde{t}_{s_2}\cos\beta_s=t_{r_1}\cos\alpha_r+\rho\tilde{t}_{r_2}\cos\beta_r,$$ (51)

and, from the condition that the sum of the traveltime of the extrapolated rays at the image point has to be equal to the traveltime of the multiple we have  

$$t_{s_2}+t_{r_2}=\tilde{t}_{s_2}+\tilde{t}_{r_2}.$$ (52)

Solving those two equations for $\tilde{t}_{s_2}$ and $\tilde{t}_{r_2}$we get

$$\begin{eqnarray} \tilde{t}_{s_2}&=&\frac{t_{r_1}\cos\alpha_r-t_{s_1}\cos\alpha_s... ...+\rho(t_{s_2}+t_{r_2})\cos\beta_s}{\rho(\cos\beta_s+\cos\beta_r)}.\end{eqnarray}$$ (53) (54)

It is interesting to check these equations in two particular cases. For a non-diffracted flat water-bottom multiple, we have $\alpha_s=\alpha_r$,$\beta_s=\beta_r$, t_s1=t_s2=t_r2=t_r1 and therefore we get $\tilde{t}_{s_2}=t_{s_2}$ and $\tilde{t}_{r_2}=t_{r_2}$ as the geometry of the problem requires. Notice that this is true for any $\rho$. The second case is for a non-diffracted water-bottom multiple migrated with water velocity. In that case, $\beta_s=\alpha_s$ and $\beta_r=\alpha_r$. Furthermore, since the multiple behaves as a primary, $(t_{s_1}+t_{s_2})\cos\alpha_s=(t_{r_1}+t_{r_2})\cos\alpha_r$ and we again get $\tilde{t}_{s_2}=t_{s_2}$ and $\tilde{t}_{r_2}=t_{r_2}$.B