Tangent to the impulse response

Following the demonstration done by Biondi (2005), the derivative of the depth with respect to the subsurface offset, at a constant image point, and the derivative of the depth with respect to the image point, at a constant subsurface offset are given by the following:

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{m_\x... ... m_\xi}{\partial \gamma} \frac{\partial h_\xi}{\partial \alpha} },\end{eqnarray}$$ (19)

and

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial m_\xi} \right\vert _{h_\x... ... m_\xi}{\partial \gamma} \frac{\partial h_\xi}{\partial \alpha} },\end{eqnarray}$$ (20)

where the partial derivatives are:

$$\begin{eqnarray} \frac{\partial z_\xi}{\partial \alpha} &=& -\frac{L}{\cos{\alph... ...\frac{(S_r-S_s)\tan{\alpha}\sin{\gamma}}{\cos^2{\gamma}} \right ].\end{eqnarray}$$ (21)

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 $\phi$ value of 2. The left panel shows the solution for equation (19). The right panel shows the solution for equation (20). 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.

 

ang_cwv_wei_surf
Figure 10 Validation of the analytical solutions for the tangent to the impulse response, the surface represents the analytical solutions and superimpose is the cut with the numerical derivative. Left: For equation (19). Right: For equation (20) analytical solutions for the tangent of the spreading surface for different values of $\phi$

B This section proofs the equivalence between the parametric equations 14, that is a direct result of trigonometry and geometry on the Figure , with the parametric equations 17, which the same equations presented by previous authors.

It is important to note that even though both parametric equations 14 and 17 are equivalent, the difference relies on the conceptual definitions of the angles involved.

The proof of this section is pure trigonemetry, and the reader can safely skip this entire Appendix. However, this Appendix is here to show that the use of our equations is valid.

First, we rewrite $\beta_s$ and $\beta_r$ as function of $\alpha$ and $\gamma$ by simple algebraic manipulation of equations 3.

$$\beta_s=\alpha - \gamma, \;\;\;\;\;\ {\rm and} \;\;\;\;\;\ \beta_r=\alpha + \gamma.$$ (22)

The first proof is the first parametric equation:

$$\begin{eqnarray} z_\xi&=& (L_s+L_r)\frac{\cos{\beta_r} \cos{\beta_s}}{\cos{\beta... ...cos^2{\alpha}- \sin^2{\gamma}}{\cos{\alpha}\cos{\gamma}} \nonumber\end{eqnarray}$$

The second parametric equation is:

$$\begin{eqnarray} 2h_\xi&=& 2h_D + (L_s+L_r)\frac{\sin{\beta_s}\cos{\beta_r}-\sin... ...\ &=& 2h_D - (L_s+L_r)\frac{\sin{\gamma}}{\cos{\alpha}} \nonumber\end{eqnarray}$$

The third parametric equation is:

$$\begin{eqnarray} m_\xi&=& m_D - \frac{(L_s+L_r)}{2}\frac{\sin{\beta_s}\cos{\beta... ...D - \frac{(L_s+L_r)}{2}\frac{\sin{\alpha}}{\cos{\gamma}} \nonumber\end{eqnarray}$$