PS-CAM theoretical impulse response

  This appendix derives the exact solution for the PS common-azimuth downward-continuation operator using the point-scatterer geometry The equation for the total travel time is the sum of a downgoing travel path with P-velocity (v_p) and an upgoing travel path with S-velocity (v_s),

 

$$t_D=\frac{\sqrt{z_\xi^2+\Vert{\bf s} -\xi_{{\bf {xy}}}\Vert^... ...rac{\sqrt{z_\xi^2+\Vert{\bf g} -\xi_{{\bf {xy}}}\Vert^2}}{v_s},$$ (97)

where $\bf s$ and $\bf g$ are the source and receiver vector locations, respectively. The vector $\xi_{{\bf {xy}}}$ denotes the point-scatterer location. The same expression can be represented in midpoint-offset coordinates (${\bf {m}},{\bf {h}}$)

$$t_D = \frac{\sqrt{z_\xi^2 + \Vert{\xi_{{\bf {xy}}}- {\bf {m}... ..._\xi^2 + \Vert{\xi_{{\bf {xy}}}- {\bf {m}}- H_X}\Vert^2}}{v_p}.$$ (98)

Figure  shows the single-mode summation surface in midpoint-offset coordinates for a point scatterer at a depth of 500 m, a P-velocity of 3000 m/s, and inline-offset of 3000 m. Figure  shows the equivalent surface for converted waves, with an S-velocity of 1500 m/s. summation

 

ppcheops
Figure 3 Single-mode (PP) total-travel-time surface (equation ) for a point scatterer at 500 m of depth in a medium with constant P-velocity=3000 m/s, and for an inline-offset=3000 m.

 

pscheops
Figure 4 Converted-mode (PS) total-travel-time surface (equation ) for a point scatterer at 500 m of depth in a medium with constant P-velocity=3000 m/s, and constant S-velocity=1500m/s, for an inline-offset=3000 m.

The following procedure shows how to go from $t_D(z_\xi,{\bf {m}},{\bf {h}})$ to $z_\xi(t_D,{\bf {m}},{\bf {h}})$:

 

$$t_D v_p = \gamma \sqrt{z_\xi^2 + \Vert{\xi_{{\bf {xy}}}- {\b... ...sqrt{z_\xi^2 + \Vert{\xi_{{\bf {xy}}}- {\bf {m}}- H_X}\Vert^2},$$ (99)

where $\gamma$ represents the P-to-S velocities ratio. If we make the following definitions,

$$\begin{eqnarray} 2 A &=& t_D v_p, \nonumber\ \alpha &=& z_\xi+ \Vert{\xi_{{\bf ... ... \beta &=& z_\xi+ \Vert{\xi_{{\bf {xy}}}- {\bf {m}}- H_X}\Vert^2, \end{eqnarray}$$ (100)

() becomes

$$2 A = \gamma \sqrt{\alpha} + \sqrt{\beta}.$$ (101)

We square both sides to get a new equation with only one square root:

$$4 A^2 - (\gamma^2 \alpha + \beta) = 2\gamma \sqrt{\alpha \beta}.$$ (102)

Squaring again to eliminate the square root, and combining elements, we obtain

$$16 A^4 - 8 A^2 (\gamma^2 \alpha + \beta) + (\gamma^2 \alpha - \beta)^2 = 0.$$ (103)

This expression is a 4^th degree polynomial in $z_\xi$; which is:

$$\begin{eqnarray} 0 &=& 16 A^4 - 8A^2 ((\gamma^2+1)z_\xi^2 + \gamma^2 (\Vert{\xi_... ...Vert^2)^2 - (\Vert{\xi_{{\bf {xy}}}- {\bf {m}}- H_X}\Vert^2)^2)^2.\end{eqnarray}$$ (104)

This can also be writen as follows

$$\begin{eqnarray} 0 &=& (\gamma^2-1)^2 z_\xi^4 \nonumber\ &+& (2 \gamma^2 (\Ver... ...\Vert^2)^2 - (\Vert{\xi_{{\bf {xy}}}- {\bf {m}}- H_X}\Vert^2)^2)^2\end{eqnarray}$$ (105)

This polynomial equation has 4 solutions, which take the following well known form:

$$z_\xi= \pm \sqrt{\frac{-b \pm \sqrt{b^2 - 4a c}}{2 a}},$$ (106)

where

$$\begin{eqnarray} a &=& \left ( \gamma^2 -1 \right )^2, \nonumber\ b &=& 2\gamma... ...Vert^2)^2 - (\Vert{\xi_{{\bf {xy}}}- {\bf {m}}- H_X}\Vert^2)^2)^2.\end{eqnarray}$$ (107)

Figure  presents the four solutions described in the previous equation. This plot, eventhough is a simple way to verify the appropiate solution for our case, it's a simple way to select the solution that is adequate to our problem. The plot suggests that the fourth solution, that is the solution with the negative both inside and outside the external square root is the appropiate solution.

 

ps4sol
Figure 5 Solutions for the analytical solution of the PS common-azimuth migration operator. The final solution is the first plot, that corresponds to the solution with both negative signs.