Wave-equation imaging

A prestack image provides information on both velocity errors and rock-property characteristics. This paper obtains the prestack image through wave-equation methods. Several authors have described this process in general, so it will not be the main focus of this paper. However, this section describes the basics of wave-equation imaging, and outlines the method we use to obtain subsurface offset-domain common-image gathers.

Biondi (2003) shows the equivalence of wave-equation source-receiver migration with wave-equation shot-profile migration. The main contribution of this paper is independent of the migration algorithm implemented, as long as the migration algorithm is based on the wavefield downward continuation, and the final prestack image is a function of the horizontal subsurface offset. For the purposes of this paper, we are using shot-profile migration as our imaging algorithm.

The final prestack image is obtained with the following imaging condition Sava and Fomel (2005):

$$I({\bf m_\xi},{\bf h_\xi}) = \sum_{\omega} U^s_z({\bf m_\xi}... ...\xi},\omega) \overline{U^r_z}({\bf m_\xi}+{\bf h_\xi},\omega).$$ (1)

Here, ${\bf {m_\xi}}=(m_{\xi},z_{\xi})$ is a vector describing the locations of the image points, and ${\bf {h_\xi}}=(h_{x_\xi},h_{z_\xi})$is a vector describing the subsurface offset. For 2-D converted-wave seismic data, the component $m_\xi$represents the horizontal coordinate, $z_\xi$ is the depth coordinate of the image point relative to a reference coordinate system, and $h_{x_\xi}$is the horizontal subsurface offset Rickett and Sava (2002). The summation over temporal frequencies ($\omega$) extracts the image $I({\bf m_\xi},{\bf h_\xi})$ at zero-time. The propagation of the receiver wavefield (U^r_z) and the source wavefield (U^s_z) is done by downward continuing the recorded data, and the given source wavelet, each one respectively as:

$$\begin{eqnarray} U^s_z & = & U^s_{z=0} e^{+iz\sqrt{\frac{\omega^2}{\vp2}-k_{m_{x... ...= & U^r_{z=0} e^{-iz\sqrt{\frac{\omega^2}{\vs2}-k_{m_{x_\xi}}^2}},\end{eqnarray}$$ (2)

where v_p is the P-wave velocity and v_s is the S-wave velocity.

The next section describes the main focus of this paper, which is the transformation of subsurface offset into the angle domain.