Image-space wave-equation tomography in the generalized source domain

Data-space encoded wavefields

The data-space encoded shot gathers are obtained by linear combination of the original shot gathers after phase encoding. For simplicity, we mainly consider plane-wave phase-encoding (Duquet and Lailly, 2006; Whitmore, 1995; Liu et al., 2006; Zhang et al., 2005) and random phase-encoding (Romero et al., 2000). Because of the linearity of the one-way wave equation with respect to the wavefield, the encoded source and receiver wavefields also satisfy the same one-way wave equations defined by Equations 13 and 14, but with different boundary conditions:

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...lta({\bf x}-{\bf x}_s) \alpha({\bf x}_s,{\bf p}_s,\omega)} \end{array} \right.,$$ (A-21)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...y,z=0,{\bf x}_s,\omega) \alpha({\bf x}_s,{\bf p}_s,\omega) \end{array} \right.,$$ (A-22)

where ${\widetilde D}({\bf x},{\bf p}_s,\omega)$ and ${\widetilde U}({\bf x},{\bf p}_s,\omega)$ are the encoded source and receiver wavefields respectively, and $\alpha({\bf x}_s,{\bf p}_s,\omega)$ is the phase-encoding function. In the case of plane-wave phase encoding, $\alpha({\bf x}_s,{\bf p}_s,\omega)$ is defined as

$$\alpha({\bf x}_s,{\bf p}_s,\omega) = e^{i\omega{\bf p}_s{\bf x}_s},$$ (A-23)

where ${\bf p}_s$ is the ray parameter for the source plane waves on the surface. In the case of random phase encoding, the phase function is

$$\alpha({\bf x}_s,{\bf p}_s,\omega) = e^{i\gamma({\bf x}_s,{\bf p}_s,\omega)},$$ (A-24)

where $\gamma({\bf x}_s,{\bf p}_s,\omega)$ is a random sequence in ${\bf x}_s$ and $\omega$. The parameter ${\bf p}_s$ defines the index of different realizations of the random sequence (Tang, 2008). The final image is obtained by applying the cross-correlation imaging condition and summing the images for all ${\bf p}_s$'s:

$$I_{\rm de}({\bf x},{\bf h}) = \sum_{{\bf p}_s}\sum_{\omega} \vert... ... x}-{\bf h},{\bf p}_s,\omega) {\widetilde U}({\bf x}+{\bf h},{\bf p}_s,\omega),$$ (A-25)

where $c=\omega$ for plane-wave phase encoding and $c=1$ for random phase encoding (Tang, 2008). It has been shown by Etgen (2005) and Liu et al. (2006) that plane-wave phase-encoding migration, by stacking a considerable number of ${\bf p}_s$, produces a migrated image almost identical to the shot-profile migrated image. If the original shots are well sampled, the number of plane waves required for migration is generally much smaller than the number of the original shot gathers (Etgen, 2005). Therefore plane-wave source migration is widely used in practice. Random-phase encoding migration is also an efficient tool, but the random phase function is not very effective in attenuating the crosstalk, especially when many sources are simultaneously encoded (Tang, 2008; Romero et al., 2000). Nevertheless, if many realizations of the random sequences are used, the final stacked image would also be approximately the same as the shot-profile migrated image. Therefore, the following relation approximately holds:

$$I({\bf x},{\bf h}) \approx I_{\rm de}({\bf x},{\bf h}).$$ (A-26)

That is, with the data-space encoded gathers, we obtain an image similar to that computed by the more expensive shot-profile migration. From Equation 25, the perturbed image can be easily obtained as follows:

$$\Delta I_{\rm de}({\bf x},{\bf h}) = \sum_{{\bf p}_s}\sum_{\omega}\vert c\vert^2 \left( \Delta \wideti... ...}_s,\omega) \widehat{\widetilde{U}}({\bf x}+{\bf h},{\bf p}_s,\omega) + \right.$$

$$\left. \widehat{\widetilde{D}}({\bf x}-{\bf h},{\bf p}_s,\omega) \Delta \widetilde{U}({\bf x}+{\bf h},{\bf p}_s,\omega) \right),$$ (A-27)

where $\widehat{\widetilde{D}}({\bf x},{\bf p}_s,\omega)$ and $\widehat{\widetilde{U}}({\bf x},{\bf p}_s,\omega)$ are the data-space encoded background source and receiver wavefields; $\Delta \widetilde{D}({\bf x},{\bf p}_s,\omega)$ and $\Delta \widetilde{U}({\bf x},{\bf p}_s,\omega)$ are the perturbed source and receiver wavefields in the data-space phase-encoding domain, which satisfy the perturbed one-way wave equations defined by Equations 17 and 18. The tomographic operator ${\bf T}$ and its adjoint ${\bf T}^{\prime }$ can be implemented in a manner similar to that discussed in Appendices B and C by replacing the original wavefields with the data-space phase encoded wavefields.

Image-space wave-equation tomography in the generalized source domain