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

the tomographic operator in the shot-profile domain

For the conventional shot-profile migration, both source and receiver wavefields are downward continued with the following one-way wave equations (Claerbout, 1971):

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...,\omega) = \overline{f_s(\omega)\delta({\bf x}-{\bf x}_s)} \end{array} \right.,$$ (A-13)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ... U(x,y,z=0,{\bf x}_s,\omega) = Q(x,y,z=0,{\bf x}_s,\omega) \end{array} \right.,$$ (A-14)

where the overline stands for complex conjugate; $D({\bf x},{\bf x}_s,\omega)$ is the source wavefield for a single frequency $\omega$ at image point ${\bf x}=(x,y,z)$ with the source located at ${\bf x}_s=(x_s,y_s,0)$; $U({\bf x},{\bf x}_s,\omega)$ is the receiver wavefield for a single frequency $\omega$ at image point ${\bf x}$ for the source located at ${\bf x}_s$; $s({\bf x})$ is the slowness at ${\bf x}$; ${\bf k}=(k_x,k_y)$ is the spatial wavenumber vector; $f_s(\omega)$ is the frequency dependent source signature, and $\overline{f_s(\omega)\delta({\bf x}-{\bf x}_s)}$ defines the point source function at ${\bf x}_s$, which serves as the boundary condition of Equation 13. $Q(x,y,z=0,{\bf x}_s,\omega)$ is the recorded shot gather for the shot located at ${\bf x}_s$, which serves as the boundary condition of Equation 14. To produce the image, the following cross-correlation imaging condition is used:

$$I({\bf x},{\bf h}) = \sum_{{\bf x}_s}\sum_{\omega} D({\bf x}-{\bf h},{\bf x}_s,\omega) U({\bf x}+{\bf h},{\bf x}_s,\omega),$$ (A-15)

where ${\bf h}=(h_x,h_y,h_z)$ is the subsurface half offset.

The perturbed image can be derived by a simple application of the chain rule to Equation 15:

$$\Delta I({\bf x},{\bf h}) = \sum_{{\bf x}_s}\sum_{\omega} \left( \Delta D({\bf x}-{\bf h},{\bf x}_s,\omega) {\widehat U}({\bf x}+{\bf h},{\bf x}_s,\omega) + \right.$$

$$\left. {\widehat D}({\bf x}-{\bf h},{\bf x}_s,\omega) \Delta U ({\bf x}+{\bf h},{\bf x}_s,\omega) \right),$$ (A-16)

where ${\widehat D}({\bf x}-{\bf h},{\bf x}_s,\omega)$ and ${\widehat U}({\bf x}+{\bf h},{\bf x}_s,\omega)$ are the background source and receiver wavefields computed with the background slowness ${\widehat s}({\bf x})$; $\Delta D({\bf x}-{\bf h},{\bf x}_s,\omega)$ and $\Delta U({\bf x}+{\bf h},{\bf x}_s,\omega)$ are the perturbed source wavefield and perturbed receiver wavefield, which are the results of the slowness perturbation $\Delta s({\bf x})$. The perturbed source and receiver wavefields satisfy the following one-way wave equations, which are linearized with respect to slowness (see Appendix A for derivations):

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...f x}_s,\omega) \\ \Delta D(x,y,z=0,{\bf x}_s,\omega) = 0 \end{array} \right. ,$$ (A-17)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...bf x}_s,\omega) \\ \Delta U(x,y,z=0,{\bf x}_s,\omega) = 0 \end{array} \right..$$ (A-18)

Recursively solving Equations 17 and 18 gives us the perturbed source and receiver wavefields. The perturbed source and receiver wavefields are then used in Equation 16 to generate the perturbed image $\Delta I({\bf x},{\bf h})$, where the background source and receiver wavefields are precomputed by recursively solving Equations 13 and 14 with a background slowness ${\widehat s}(\bf x)$. Appendix B gives a more detailed matrix representation of how to evaluate the forward tomographic operator ${\bf T}$.

To evaluate the adjoint tomographic operator ${\bf T}'$, we first apply the adjoint of the imaging condition in Equation 16 to get the perturbed source and receiver wavefields $\Delta D({\bf x},{\bf x}_s,\omega)$ and $\Delta U({\bf x},{\bf x}_s,\omega)$ as follows:

$$\Delta D({\bf x},{\bf x}_s,\omega) = \sum_{\bf h} \Delta I({\bf x},{\bf h})\overline{{\widehat U}({\bf x}+{\bf h},{\bf x}_s,\omega)},$$ (A-19)

$$\Delta U({\bf x},{\bf x}_s,\omega) = \sum_{\bf h} \Delta I({\bf x},{\bf h})\overline{{\widehat D}({\bf x}-{\bf h},{\bf x}_s,\omega)}.$$ (A-20)

Then we solve the adjoint equations of Equations 17 and 18 to get the slowness perturbation $\Delta s({\bf x})$. Again, in order to solve the adjoint equations of Equations 17 and 18, the background source wavefield ${\widehat D}({\bf x},{\bf x}_s,{\omega})$ and the background receiver wavefield ${\widehat U}({\bf x},{\bf x}_s,\omega)$ have to be computed in advance. Appendix C gives a more detailed matrix representation of how to evaluate the adjoint tomographic operator ${\bf T}^{'}$.

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