Traveltime of refracted rays

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

$$t_{s_1}\cos\alpha_s+\rho\tilde{t}_{s_2}\cos\beta_s=t_{r_1}\cos\alpha_r+\rho\tilde{t}_{r_2}\cos\beta_r,$$ (1)

and, from the imaging condition (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

$$t_{s_2}+t_{r_2}=\tilde{t}_{s_2}+\tilde{t}_{r_2}.$$ (2)

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

$$\tilde{t}_{s_2} = \frac{t_{r_1}\cos\alpha_r-t_{s_1}\cos\alpha_s+\rho(t_{s_2}+t_{r_2})\cos\beta_r}{\rho(\cos\beta_s+\cos\beta_r)},$$ (3)

$$\tilde{t}_{r_2} = \frac{t_{s_1}\cos\alpha_s-t_{r_1}\cos\alpha_r+\rho(t_{s_2}+t_{r_2})\cos\beta_s}{\rho(\cos\beta_s+\cos\beta_r)}.$$ (4)

It is interesting to check these equations in two particular cases:

1. For a specular multiple from a flat water-bottom, we have $\alpha_s=\alpha_r$, $\beta_s=\beta_r$, $t_{s_1}=t_{s_2}=t_{r_2}=t_{r_1}$ 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 $\rho$.
2. For a specular water-bottom multiple migrated with water velocity ($\rho=1$), we have $\beta_s=\alpha_s$ 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 $\tilde{t}_{r_2}=t_{r_2}$.