Traveltime computations for dipping water-bottom multiple

In this Appendix I derive equations 39-45.

 

mul_sktch9 Figure 21 Sketch to show the computation of ts1 and ts2 for a non-diffracted multiple from a dipping water-bottom.

From triangle ABC in Figure  we immediately get

$$t_{s_1}=\frac{\tilde{Z}_s}{\cos(\alpha_s+\varphi)}$$ (57)

and applying the law of sines to triangle ACD we get

$$t_{s_2}=\frac{t_{s_1}\cos\alpha_s}{\cos(\alpha_s+2\varphi)}=... ...cos\alpha_s}{V_1\cos(\alpha_s+\varphi)\cos(\alpha_s+2\varphi)}.$$ (58)

 

mul_sktch10 Figure 22 Sketch to show the computation of tr2 and tr1 for a non-diffracted multiple from a dipping water-bottom.

Similarly, repeated application of the law of sines to triangles CDE and DEF in Figure  gives

$$\begin{eqnarray} t_{r_2}&=&\frac{t_{s_2}\cos(\alpha_s+\varphi)}{\cos(\alpha_s+3\... ...s\cos\alpha_s}{V_1\cos(\alpha_s+3\varphi)\cos(\alpha_s+4\varphi)}.\end{eqnarray}$$ (59) (60)

 

mul_sktch11 Figure 23 Sketch to show the computation of $\tilde{Z}_s$ in equation 61.

These equations are in terms of $\tilde{Z}_s$, which is not known. However, from Figure  we see that  

$$\tilde{Z}_s=\tilde{Z}_D-h_D\sin\varphi,$$ (61)

and $\tilde{Z}_D$ can be computed from the traveltime of the zero surface-offset trace, since, according to Figure 

$$t_m(0)=\frac{2\tilde{Z}_D}{V_1\cos\varphi}+\frac{2\tilde{Z}_... ...\varphi}=\frac{2\tilde{Z}_D(1+\cos(2\varphi))}{V_1\cos\varphi},$$ (62)

from which it follows immediately that  

$$\tilde{Z}_D=\frac{V_1t_m(0)\cos\varphi}{2[1+\cos(2\varphi)]}.$$ (63)

 

mul_sktch12 Figure 24 Sketch to show the computation of $\tilde{Z}_D$ in equation 63.

Finally, we need to compute $\alpha_s$. Applying the law of sines to triangle ABC in Figure  we get

$$\sin(\alpha_s+2\varphi)=\frac{2h_D\cos(2\varphi)}{V_1t_m},$$ (64)

from which we get

$$\alpha_s=\sin^{-1}\left[\frac{2h_D\cos(2\varphi)}{V_1t_m}\right]-2\varphi.$$ (65)

 

mul_sktch16 Figure 25 Sketch to compute the takeoff angle of the source ray from a non-diffracted multiple from a dipping water-bottom.

D