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Theory of daylight imaging

We will now describe how to image either the source distribution or the reflectivity distribution from passive seismic data in a v(x,y,z) medium. The sources are assumed to be distributed anywhere in space, and the time histories of each source are assumed to be uncorrelated. Neither the source location or time history are known. Without loss of generality we conveniently assume one source and one scatterer, but the resulting migration formula are also applicable to multiple sources and scatterers.

For an earth model with a free-surface, a smoothly varying velocity distribution and a single point scatterer at xo, the wavefield, $W(\vec r_{g'}\vert\vec r_s,\omega )$, recorded at ${\vec r_{g'}}$ due a source at ${\vec r_{s}}$ is given by
   \begin{eqnarray}
W(\vec r_{g'}\vert\vec r_s,\omega) & \approx & 
G(\vec r_{g'}\v...
 ... }+ \tau_{g^{''}x_o }+ \tau_{x_og'})} ]
\; F(\omega), \nonumber\ \end{eqnarray}
where $G(\vec{r}_g\vert\vec{r}_s,\omega)$ is the WKBJ Green's function for an impulsive source at ${\vec r_{s}}$ and a geophone at $\vec{r}_g$, and $F(\omega)$ is the complex source spectrum.

The first term on the RHS represents the direct arrival (see top figure in Figure 1), the second term represents the scattered field excited by the direct arrival, and the third term represents the scattered arrival that is generated by a specular free-surface reflection (see middle figure in Figure 1). Here, R is the scattering coefficient, $\tau_{gg'}$ is a solution to the eikonal equation for a source at g and a receiver at g', and the geometrical-spreading factors have been harmlessly dropped. The specular-reflection point on the free surface is denoted by g'' and the location of the recording geophone is denoted by g'.

To eliminate the unknown phase of the source, we crosscorrelate the wavefield, $W(\vec r_{g'}\vert\vec r_s,\omega )$ recorded at ${\vec r_{g'}}$ with $W(\vec r_{g_o}\vert\vec r_s,\omega )$ recorded at $\vec r_{g_o}$to get
\begin{eqnarray}
W(\vec r_{g'}\vert\vec r_s,\omega ) W(\vec r_{g_o}\vert\vec r_s...
 ... G(\vec r_{g_o}\vert\vec r_s,\omega )^* \;
 \vert F(\omega)\vert^2\end{eqnarray} (1)
If we then assume source is a white source spectrum such that $\vert F(\omega)\vert^2\approx 1$, the source term can be dropped entirely, leaving
   \begin{eqnarray}
W(\vec r_{g'}\vert\vec r_s,\omega ) 
 W(\vec r_{g_o}\vert\vec r...
 ...\tau_{x_og'} -
\tau_{sg_o })}
+ {\rm\it other~terms}, \nonumber\ \end{eqnarray}
where the first expression on the RHS corresponds to the correlation of the conjugate direct wave recorded at geophone go with the direct wave at g'; and the second term corresponds to the correlation of the conjugate direct wave at go with the ghost reflection recorded at g'. The other terms correspond to the other correlations which will not be needed for imaging, and are presumed to cancel upon migration. This last assumption about cancellation is similar to the standard assumption in migration, i.e., all migrated arrivals incoherently superimpose except those that are tuned to the specified imaging condition. For the above equation, we will tune the imaging conditions so that either the ${\rm\it Direct}_{g_o}^* ~{\rm\it Direct}_{g'}$terms are migrated to image the source distribution, or the ${\rm\it Direct}_{g_o}^*~{\rm\it Ghost}_{g'}$ terms are migrated to image the reflectivity distribution.



 
next up previous print clean
Next: Source location imaging Up: Schuster & Rickett: Daylight Previous: Extending daylight imaging
Stanford Exploration Project
9/5/2000