Tangent to the impulse response

Following Biondi's 2005 demostration, the derivative of the image depth ($z_\xi$) with respect to the subsurface-offset ($h_\xi$), at a constant image midpoint ($m_\xi$), and the derivative of the depth with respect to the image point, at a constant subsurface offset are given by the following:

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{m_\x... ... m_\xi}{\partial \gamma} \frac{\partial h_\xi}{\partial \alpha} },\end{eqnarray}$$ (84)

and

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial m_\xi} \right\vert _{h_\x... ... m_\xi}{\partial \gamma} \frac{\partial h_\xi}{\partial \alpha} },\end{eqnarray}$$ (85)

where the partial derivatives are:

$$\begin{eqnarray} \frac{\partial z_\xi}{\partial \alpha} &=& -\frac{L}{\cos{\alph... ...\frac{(S_r-S_s)\tan{\alpha}\sin{\gamma}}{\cos^2{\gamma}} \right ].\end{eqnarray}$$ (86)

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 . The right panel shows the solution for equation . 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 (Appendix A).

 

ang_cwv_wei_surf
Figure 2 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 . Right: For equation analytical solutions for the tangent of the spreading surface for different values of $\phi$