GROUP-DOMAIN ALGORITHMS

With the help of the associated ray equation, a Kirchhoff migration in x-t domain can be implemented directly or traveltimes can be calculated from the dispersion relation using anisotropic finite difference traveltimes. Triplications are caused by concavities in the phase slowness surface. The double elliptic approximation can model concavities well; thus triplications are produced. When the ray equation instead of the dispersion relation is used, the group velocity surface is approximated instead of the phase velocity surface. The approximation of the group velocity surface fails to follow the triplication. This is the fundamental difference between the $\omega-k$ and x-t methods. Typical x-t methods is Kirchhoff modeling and migration. In order to be accurate, such an algorithm has to handle triplications properly, which in general is not as easy in x-t as in $\omega-k$. When the double elliptic form is used, the exact curve cannot be approximated well in the area of triplications. An algorithm which uses the group domain approximation (see Figure , middle bottom) is not able to produce triplications.