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Next: Case 2: Postack Imaging. Up: Waveform Inversion Algorithm Previous: Waveform Inversion Algorithm

Case 1: Prestack Migration.

Assume a wideband source, so $\tilde W(\omega)=1$, and high frequencies so that the asymptotic Green's function can be used:

 
$\displaystyle \tilde G(\bf {r}\vert\bf {r}_s)$ = $\displaystyle \frac{e^{i \omega \tau_{sr}}}{r},$ (3.7)

where $\tau_{sr}$ is the time for energy to propagate from the source at $\bf {r}_s$to the interrogation point at $\bf {r}$, and $1/r_{sr} =1/\vert\bf {r}-\bf {r}_s\vert$ is the geometrical spreading term. Here we assume that the forward Fourier transform kernel is $e^{i\omega t}$while the kernel for the inverse Fourier transform is $e^{-\omega t}$.
  
Figure: Single shot reflection gather for a point scatterer model with a background velocity of v0. The moveout curve $\tau (x) = \tau _{gr}+\tau _{sr}$ denoted by the dashed line is that of a shifted hyperbola, where $\tau _{sr} = d/v_0$ is the propagation time from the source to the scatterer and $\tau _{sr} = [(d/v_0)^2 + (x/v_0)^2]^{1/2}$ is the propagation time from the scatterer to the geophone.
\begin{figure}\centering
\psfig{figure=b.ps,width=5.0in,height=3.0in}\psfig{figure=a.ps,width=5.0in,height=3.0in}\end{figure}

Assuming a single source and an homogeneous model with a buried scatterer (see Figure 3.3), and substituting equation 3.8 into equation 3.6 yields

 
$\displaystyle \gamma (\bf {r})$ = $\displaystyle 2 s(\bf {r}) \sum_{\omega} \sum_g \Delta
\tilde P(\bf {r}_g,\bf {r}_s)^* \omega^2
e^{i \omega (\tau_{sr}+\tau_{gr} ) }/( r_{sr}r_{gr} ),$  
  = $\displaystyle 2 s(\bf {r}) \sum_{\omega} \sum_g [\Delta
\tilde P(\bf {r}_g,\bf {r}_s) \omega^2
e^{-i \omega (\tau_{sr}+\tau_{gr} )} ]^* /( r_{sr}r_{gr} ),$  
  $\textstyle \cong$ $\displaystyle - \sum_g \Delta \ddot{p}(\bf {r}_g,\bf {r}_s, \tau_{sr}+\tau_{gr} ),$ (3.8)

where $\Delta p(\bf {r}_g,\bf {r}_s,t)$ is the transform of $\Delta \tilde P(\bf {r}_g,\bf {r}_s)$, and the terms associated with geometrical spreading and $s(\bf {r})$ have been neglected in the last expression. The double time derivative sharpens up the image, which corrects for the image smoothing due to the summation over the geophone coordinates (as well as the smoothing from a possible summation over source coordinates if more than one shot gather is used).

Equation 3.9 says that $\gamma(\bf {r})$ at each image point $\bf {r}$ is computed by summing the data residuals along a hyperbola-like curve, where the dashed hyperbola shown in Figure 3.3 is denoted by $\tau=\tau_{sr}+\tau_{gr}$. Here $\tau_{sr}$ denotes the propagation time from the source to the buried scatterer, and $\tau_{gr}$ denotes the propagation time from the scatterer to the geophone at the surface. Equation 3.9 approximates the prestack Kirchhoff migration equation for a single source gather.

As an example, the top picture in Figure 3.4 depicts the synthetic shot gather for a point scatterer in a homogeneous medium, and the bottom picture depicts the result of applying an equation similar to that of equation 3.9. Note that the point scatterer in the bottom picture is well imaged.

  
Figure 3.4: Common shot gather (top figure) seismograms and associated (bottom figure) migrated section for a homogeneous half space model with a buried point scatterer. 100 shot gathers were used to obtain this image, with 101 traces per shot gather evenly distributed over an aperture of 10,000 feet.
\begin{figure}
\centering
\psfig{figure=ch9.fig2c.ps,height=3.0in,width=5.0in}\psfig{figure=ch9.fig2d.ps,height=3.0in,width=5.0in}\end{figure}


next up previous contents
Next: Case 2: Postack Imaging. Up: Waveform Inversion Algorithm Previous: Waveform Inversion Algorithm
Gerard Schuster
1998-07-29