Common-offset prestack time migration and data regularization

The time-distance relation for a shot-receiver pair is  

$$\sqrt{\left(x-h_{x} \right) ^{2}+\left(y-h_{y} \right) ^{2}+... ...(x+h_{x} \right) ^{2}+\left(y+h_{y} \right) ^{2}+z^{2}}=vt_{h},$$ (9)

where t_h is the two-way traveltime of a non-zero-offset shot-receiver pair, h_x is the in-line component of the half-offset, and h_y 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:  

$$\left(\frac{\acute{x}}{a_{\acute{x}}} \right) ^{2}+\left( \f... ..._{\acute{y}}}\right) ^{2}+\left(\frac{z}{a_{z}} \right) ^{2}=1,$$ (10)

where $a_{\acute{x}}$,$a_{\acute{y}}$and a_z 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 t_n is the two-way traveltime after NMO. Equation () can be rewritten as  

$$\left(\frac{\acute{x}}{\frac{vt_{h}}{2}} \right) ^{2}+\left(... ...}}\right) ^{2}+\left(\frac{z}{\frac{vt_{n}}{2}} \right) ^{2}=1.$$ (11)

Further, equation () can be changed into  

$$\frac{\acute{x}^{2}}{1+\left( \frac{2h}{vt_{n}}\right) ^{2}}+\acute{y}^{2}+z^{2}=\left(\frac{vt_{n}}{2} \right) ^{2}.$$ (12)

Defining $\acute{X}^{2}=\frac{\acute{x}}{\sqrt{1+\left( \frac{2h}{vt_{n}}\right) ^{2}}}$ yields:  

$$\acute{X}^{2}+\acute{y}^{2}+z^{2}=\left(\frac{vt_{n}}{2} \right) ^{2}.$$ (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  

$$k_{\acute{X}}=k_{\acute{x}}\sqrt{1+\left( \frac{2h}{vt_{n}}\right) ^{2}}.$$ (14)

Therefore, the dispersion relation of equation () is  

$$\left(\frac{v}{2} \right) ^{2}\left[k_{\acute{X}}^{2} +k_{\acute{y}}^{2}+k_{z}^{2}\right]=\omega_{n}^{2} .$$ (15)

Substituting () into the above formula yields  

$$\left(\frac{v}{2} \right) ^{2}\left[k_{\acute{x}}^{2}\left( ... ...{2} \right) +k_{\acute{y}}^{2}+k_{z}^{2}\right]=\omega_{n}^{2},$$ (16)

which can be rewritten as follows:  

$$k_{z}=-sgn\left(\omega_{n} \right) \sqrt{\left( \frac{2\omeg... ...eft( \frac{2h}{vt_{n}}\right) ^{2}\right) k_{x}^{2}-k_{y}^{2}}.$$ (17)

This is the dispersion relation of the common-offset prestack migration equation. In the time domain, the dispersion relation is  

$$k_{\tau}=-sgn\left(\omega_{n}\right) \sqrt{\omega_{n}^{2}-\l... ...\right) t_{n}}\right) ^{2}\right) k_{x}^{2}+k_{y}^{2}\right] }.$$ (18)

Therefore, common-offset prestack time migration (PSTM) can be implemented with the following relation:  

$$\textbf{m}\left(\omega_{\tau}, k_{x},k_{y} \right)=\int dt_{... ...{n}} \textbf{d}\left(t_{n}, k_{x},k_{y},h\right)\right\rbrace .$$ (19)

The term in the braces represents the wave-field extrapolation, and the integral at t_n=0 serves to extract the imaging values. Then, the common-offset inverse PSTM is  

$$\textbf{d}\left(t_{n},k_{x},k_{y},h\right)=\int d\omega_{\ta... ...extbf{m}\left(\omega_{\tau}, k_{x},k_{y} \right)\right\rbrace .$$ (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.