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From dip to no dip for non-diffracted multiple

In this Appendix, I show that the equations for the non-diffracted multiple from a dipping water-bottom reduce to the equations for a flat water-bottom when $\varphi=0$ as they should. Setting $\varphi=0$ in equations 39 through 42 we obtain
\begin{displaymath}
t_{s_2}=t_{r_2}=t_{r_1}=t_{s_1}=\frac{\tilde{Z}_s}{\cos\alpha_s}\end{displaymath} (66)
and from equations 5 and 6 we get (as discussed at the end of Appendix A) $\tilde{t}_{r_2}=t_{r_2}$ and $\tilde{t}_{s_2}=t_{s_2}$.Therefore,
\begin{eqnarray}
h_\xi&=&2h_D-V_1[t_{s_1}\sin\alpha_s+t_{r_1}\sin\alpha_r+\rho(\...
 ..._{s_1}\sin\alpha_s(1+\rho^2)\\ &=&2h_D-h_D(1+\rho^2)=h_D(1-\rho^2)\end{eqnarray} (67)
(68)
(69)
(70)
Similarly,
\begin{eqnarray}
z_\xi&=&V_1(t_{r_1}+\rho\tilde{t}_{s_2}\sqrt{1-\rho^2\sin^2\alp...
 ...^2(1-\rho^2)}=Z_{wb}+\frac{\rho}{2}\sqrt{Z_{wb}^2+h_D^2(1-\rho^2)}\end{eqnarray} (71)
(72)
(73)
(74)
(75)
Finally,
\begin{eqnarray}
m_\xi&=&m_D+\frac{V_1}{2}(t_{s_1}\sin\alpha_s-t_{r_1}\sin\alpha...
 ...}\sin\alpha_s)+\rho^2(t_{s_2}\sin\alpha_s-t_{s_2}\sin\alpha_s)=m_D\end{eqnarray} (76)
(77)
E
next up previous print clean
Next: Computation of takeoff angles Up: Alvarez: Multiples in image Previous: Traveltime computations for dipping
Stanford Exploration Project
11/1/2005