Depth Controlled Reflection tomography

Borehole seismic data can provide the exact position of the subsurface reflectors at the borehole. Therefore, from borehole data, we can obtain $\Delta r$ at these locations. $\Delta r$ can be backpropagated to slowness perturbation using equation (2). In order to integrate the depth control to reflection tomography effectively, we transfer the reflector movement $\Delta r$ to traveltime perturbation along normal ray, $\Delta t_n$. The traveltime perturbation $\Delta t_n$ then is backpropagate to slowness perturbation using following equation:

$$\begin{eqnarray} \Delta t_n \approx -\int_{l_n} \Delta s dl_n\end{eqnarray}$$ (7)

Combining equation (4) and (7), we can obtain a depth controlled reflection tomography (DCRT) scheme:

$$\begin{eqnarray} {\bf \Delta t \approx T_r\Delta s } \\ {\bf \Delta t_n \approx T_n \Delta s} \\ {\bf 0 \approx \epsilon A \Delta s}\end{eqnarray}$$ (8) (9) (10)

Here, fitting goal (8) and (9) correspond to equation (4) and (7), respectively. Fitting goal 10 is the model styling goal. $\bf A$ is a regularization operator. We use a Laplacian as regularization operator for the following application.