VTI migration velocity analysis using RTM |

First, for each image slice in the subsurface-offset domain , we compute a weighted image using equation 27. Then we move on to equation 25. We can rearrange the independent and commutable operators as follows:

Operator corresponds to differentiating and once reversely in time and setting , and fields to zero. Notice that the directions of propagation and differentiation in time of wavefield are the same. Therefore, we can compute the time derivative during the same process as the propagation. Then we shift the reverse-time derivative by in , and multiply it with the weighted image . This product is shifted again by . Finally, we sum over the contributions from all subsurface-offset image slices to get an effective source term . Next, we solve equation 33 for backward in time, using as the source.

At the same time, in equation 28 is a sparse matrix, with non-zero elements only for and . We can therefore write everything out explicitly:

The explicit forms for the complete gradients are:

and

VTI migration velocity analysis using RTM |

2012-05-10