Reflectivity distribution imaging

To image the unknown scatterer location and strength at $\vec x_o$, we simply identify the migration operator which, when applied to the crosscorrelated data in equation (4), cancels the phase of the ${\rm\it Direct}_{g_o}^*~{\rm\it Ghost}_{g'}$ term. Such a migration operator is given by

$$\begin{eqnarray} e^{-i\omega ( \tau_{g_o x }+ \tau_{xg'})},\end{eqnarray}$$ (3)

where x and g_o denote, respectively, the trial scatterer and specular-reflector point locations respectively. Application of this migration operator to equation (3) will annihilate the phase term of the ${\rm\it Direct}_{g_o}^* ~{\rm\it Ghost}_{g''}$ term in equation (3) when both the trial image point, $\vec x$, coincides with the actual source location denoted by $\vec x_o$, and the trial specular-reflection point, $\vec r_{g_o}$, coincides with the actual specular-reflection point, $\vec r_{g''}$. In other words, we require both

$$\begin{eqnarray} \vec x & \rightarrow& x_o, ~{\rm and} \ \vec r_{g_o} & \rightarrow & \vec r_{g''}.\end{eqnarray}$$ (4) (5)

This can be understood more clearly by noting that the migrated reflectivity section is given by summation of the migrated crosscorrelation data over all geophone pairs

$$\begin{eqnarray} m_{\rm refl}(\vec x, \omega) &=& \sum_{g_o} \sum_{g'} \; W(\vec... ...x }- \tau_{xg'})} + {\rm\it all~other~terms} \right]. \nonumber\ \end{eqnarray}$$

As $\vec x \rightarrow x_o$ and as $\vec r_{g_o} \rightarrow \vec r_{g''}$, then it is clear that the above phase goes to zero, leading to constructive interference at the scatterer's location. In practice, these two conditions will be fulfilled if the specular-reflection point is within the aperture of the recording array and the geophone array is sufficiently dense.

 

fig2
Figure 2 Ghost reflection rays associated with a source at s, a trial image point at x, and a scatterer at x_o.

 

fig3
Figure 3 As x approaches x₀ and g_o approaches g'' in Figure 2, the dashed rays coincide with the solid rays. Thus the Direct Ghost correlation term will have zero phase, leading to maximal constructive interference of the migration section at the actual scatterer point.