Source location imaging

To image the unknown source location at ${\vec r_{s}}$ from the data given in equation (3), we simply identify the migration operator which when applied to the above equation cancels the phase of the ${\rm\it Direct}_{g_o}^* ~{\rm\it Direct}_{g'}$ term. Such a migration operator is given by

$$\begin{eqnarray} e^{ -i\omega (\tau_{s'g' }- \tau_{s'g_o })} ,\end{eqnarray}$$ (2)

where s' denotes the trial source-point location. Application of this migration operator to the crosscorrelated data in equation (3) will annihilate the phase of the ${\rm\it Direct}_{g_o}^* ~{\rm\it Direct}_{g'}$ term when the trial image point $\vec r_{s'}$ coincides with the actual source location denoted by ${\vec r_{s}}$.The migration section is then given by summation over all geophone pairs

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

The migration operator in equation (5) is "tuned" to image the source location so that as $\vec r_{s'}\rightarrow\vec r_{s}$, the ${\rm\it Direct}_{o}^* ~{\rm\it Direct}_{g'}$ term will constructively interfere while the all other terms tend to cancel.

 

fig1
Figure 1 (Top) Direct ray and a (middle) scattered ray excited by a specular free-surface reflection associated with a source at s and a scatterer at x_o. Bottom figure denotes the rays associated with the migration operator for free-surface reflections.