Slowness update and the Born approximation

We generate the image perturbation by taking the difference between a reference image and an improved version of it. An underlying assumption of the inversion process is that we are within the Born approximation, which requires that the wavelets with which we operate are not more than $\pi/4$ out of phase. What happens if this condition is not fulfilled? In other words, what can we do if the enhanced image is so different from the original image that the wavelets we subtract are not in-phase anymore?

A possible answer to this question, although not necessarily the only one, is that we need to be conservative at the time we generate the improved image. For this, we can scale the velocity-ratio parameter surface Sava (1999), which controls the amount of enhancement in the image closer to unity, that is, closer to the original image (Figure 3).

 

ratscale Figure 3 An illustration of velocity ratio scaling which makes the Born approximation valid. (a) is a gather extracted from the original image ($\gamma=1$). (c) is the same gather after residual migration with the correct ratio. (b) is the same gather after residual migration with a scaled ratio. Although the ideal image is represented by (c), we cannot use this image because the wavelets are not within the Born approximation. Instead, we use (b), which is within the Born approximation and indicates the same direction of image improvement as the optimal ratio.

The shortcoming of this procedure is that we reduce the magnitude of image perturbation, although we preserve a more important parameter - its direction. The scaled-down restored image falls within the limits of the Born approximation with respect to the original; therefore, we can safely invert for the slowness perturbation. However, the slowness perturbation we obtain depends on the scaling we have done on the images, although it has the correct direction.

Next we need to scale the slowness perturbation back up to the value corresponding to the correct velocity ratio measured from the residual migrated images. So, how do we do this? A possible solution is to run a line search using the slowness perturbation we have inverted, with the goal of maximizing the energy of the migrated image. Mathematically, this goal can be expressed as  

$$\max_{\alpha} \Vert \mathcal L\left(\bf s+ \alpha\Delta \bf s\right)\Vert,$$ (3)

where $\alpha$ is the scaling factor for the model, and $\bf s$ is the background slowness model.