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Common-offset prestack time migration and data regularization

The time-distance relation for a shot-receiver pair is  
 \begin{displaymath}
\sqrt{\left(x-h_{x} \right) ^{2}+\left(y-h_{y} \right) ^{2}+...
 ...(x+h_{x} \right) ^{2}+\left(y+h_{y} \right) ^{2}+z^{2}}=vt_{h},\end{displaymath} (9)
where th is the two-way traveltime of a non-zero-offset shot-receiver pair, hx is the in-line component of the half-offset, and hy is the cross-line component of the half-offset. For simplicity, the connection line of the shot and receiver points is parallel to the x-axis of the Cartesian coordinate system. Therefore, we have the following simple equation which delineates the isochron surface of the prestack migration:  
 \begin{displaymath}
\left(\frac{\acute{x}}{a_{\acute{x}}} \right) ^{2}+\left( \f...
 ..._{\acute{y}}}\right) ^{2}+\left(\frac{z}{a_{z}} \right) ^{2}=1,\end{displaymath} (10)
where $a_{\acute{x}}$,$a_{\acute{y}}$and az are the half-lengths of the axes of the rotary isochron ellipse in the case of constant velocity. If $\acute{y}=z=0$, then $a_{\acute{x}}=\acute{x}=\frac{vt_{h}}{2}$; If $\acute{x}=z=0$, then $a_{\acute{y}}=\acute{y}=\sqrt{\left(\frac{vt_{h}}{2} \right) ^{2}-h^{2}}=\frac{vt_{n}}{2}$; If $\acute{x}=\acute{y}=0$, then $a_{z}=z=\sqrt{\left(\frac{vt_{h}}{2} \right) ^{2}-h^{2}}=\frac{vt_{n}}{2}$. The variable tn is the two-way traveltime after NMO. Equation ([*]) can be rewritten as  
 \begin{displaymath}
\left(\frac{\acute{x}}{\frac{vt_{h}}{2}} \right) ^{2}+\left(...
 ...}}\right) ^{2}+\left(\frac{z}{\frac{vt_{n}}{2}} \right) ^{2}=1.\end{displaymath} (11)
Further, equation ([*]) can be changed into  
 \begin{displaymath}
\frac{\acute{x}^{2}}{1+\left( \frac{2h}{vt_{n}}\right) ^{2}}+\acute{y}^{2}+z^{2}=\left(\frac{vt_{n}}{2} \right) ^{2}.\end{displaymath} (12)
Defining $\acute{X}^{2}=\frac{\acute{x}}{\sqrt{1+\left( \frac{2h}{vt_{n}}\right) ^{2}}}$ yields:  
 \begin{displaymath}
\acute{X}^{2}+\acute{y}^{2}+z^{2}=\left(\frac{vt_{n}}{2} \right) ^{2}.\end{displaymath} (13)
Equation ([*]) is in the form of a poststack migration. Therefore, prestack migration can be explained as a poststack migration on a post-NMO data set. We know that  
 \begin{displaymath}
k_{\acute{X}}=k_{\acute{x}}\sqrt{1+\left( \frac{2h}{vt_{n}}\right) ^{2}}.\end{displaymath} (14)
Therefore, the dispersion relation of equation ([*]) is  
 \begin{displaymath}
\left(\frac{v}{2} \right) ^{2}\left[k_{\acute{X}}^{2} +k_{\acute{y}}^{2}+k_{z}^{2}\right]=\omega_{n}^{2} .\end{displaymath} (15)
Substituting ([*]) into the above formula yields  
 \begin{displaymath}
\left(\frac{v}{2} \right) ^{2}\left[k_{\acute{x}}^{2}\left( ...
 ...{2} \right) +k_{\acute{y}}^{2}+k_{z}^{2}\right]=\omega_{n}^{2},\end{displaymath} (16)
which can be rewritten as follows:  
 \begin{displaymath}
k_{z}=-sgn\left(\omega_{n} \right) \sqrt{\left( \frac{2\omeg...
 ...eft( \frac{2h}{vt_{n}}\right) ^{2}\right) k_{x}^{2}-k_{y}^{2}}.\end{displaymath} (17)
This is the dispersion relation of the common-offset prestack migration equation. In the time domain, the dispersion relation is  
 \begin{displaymath}
k_{\tau}=-sgn\left(\omega_{n}\right) \sqrt{\omega_{n}^{2}-\l...
 ...\right) t_{n}}\right) ^{2}\right) k_{x}^{2}+k_{y}^{2}\right] }.\end{displaymath} (18)
Therefore, common-offset prestack time migration (PSTM) can be implemented with the following relation:  
 \begin{displaymath}
\textbf{m}\left(\omega_{\tau}, k_{x},k_{y} \right)=\int dt_{...
 ...{n}} \textbf{d}\left(t_{n}, k_{x},k_{y},h\right)\right\rbrace .\end{displaymath} (19)
The term in the braces represents the wave-field extrapolation, and the integral at tn=0 serves to extract the imaging values. Then, the common-offset inverse PSTM is  
 \begin{displaymath}
\textbf{d}\left(t_{n},k_{x},k_{y},h\right)=\int d\omega_{\ta...
 ...extbf{m}\left(\omega_{\tau}, k_{x},k_{y} \right)\right\rbrace .\end{displaymath} (20)
Similarly, the term in the braces represents the wave-field extrapolation, which is an inverse migration. The integral is an inverse Fourier transform.

In the presence of moderate lateral velocity variations, prestack time migration can be expressed as follows:
\begin{eqnarray}
\textbf{m}\left(\tau, m_{x},m_{y} \right)&=&\int dx_{s} \int dx...
 ...eft( t, x_{s},x_{g}, h \right) \delta\left(t=t_{s}+t_{g} \right) ,\end{eqnarray}
(21)
where $t=t_{s}+t_{g}=\sqrt{\left(\frac{m_{x}-h_{x}}{v_{rms}} \right) ^{2}+\left(\frac{...
 ...eft(\frac{m_{y}+h_{y}}{v_{rms}} \right) ^{2}+\left( \frac{\tau}{2}\right) ^{2}}$. W1 is the amplitude weight, and $\tau$ is the two way traveltime along the imaging ray.

The inverse PSTM is
\begin{eqnarray}
\textbf{d}\left(t, x_{s},x_{g} ,h \right)&=&\int dm_{x} \int dm...
 ...x},m_{y} \right)\delta\left(t=-\left( t_{s}+t_{g}\right) \right) .\end{eqnarray}
(22)
() give a general theory of data mapping. From here, we will develop some practical approaches for data mapping.


next up previous print clean
Next: Aliasing and anti-aliasing Up: Seismic data preprocessing Previous: Seismic data preprocessing
Stanford Exploration Project
5/3/2005