Discussion

In this shortnote, we presented a method to add depth control to reflection tomography. We transfer the exact reflector movement, which can be obtained from borehole data, to the traveltime perturbation along the normal ray. The traveltime perturbation along normal ray provides another data fitting goal for reflection tomography. By simultaneously backpropagating normal ray traveltime perturbation and reflection traveltime perturbation, we can improve the inversion result.

In the finished work, we obtained the normal shift between the correct reflection point and apparent reflection point, $\Delta r$, then transfer it to the traveltime perturbation along normal ray for backpropagation. Notice in Figure , by summing $\Delta r$ and residual moveout $\Delta r_2$, we can obtain the total normal shift $\Delta r_1$, which can be transfered to traveltime perturbation along the offset ray according to equation (5). Therefore, instead of using equation (7) for backpropagation, we can backpropagate the traveltime along the offset ray by using the following linear relationship:

$$\begin{eqnarray} \Delta t_o = \int_{l_o} \Delta s dl_o.\end{eqnarray}$$ (11)

An obvious advantage of backpropagating along the offset ray is that we can obtain better ray coverage. Backpropagating along normal ray can only obtain velocity along normal ray direction, whereas, when backpropagating along offset ray, we can obtain a much wider ray coverage with varying aperture angle of offset ray.

In the completed work, we did not apply any weighting between fitting goal (8) and (9). With an appropriate weighting scheme, the DCRT should improve the inversion result.

Another way to improve DCRT result is to use a spatially-varying Lagrange multiplier $\epsilon$. We assume all the reflection points within a local area have the same normal shift. Such an approximation is more reliable for the reflection points near the borehole, and less reliable for those points away from the borehole. In order to take this into account during inversion, we can apply spatially-varying $\epsilon$ to the model styling goal (10). We can apply small $\epsilon$ for the area close to the well to emphasize the data fitting, whereas big $\epsilon$ for the area away from the well to emphasize the model styling.