The double-elliptic approximation in the group and phase domains

in the dispersion-relation domain

There is no such problem in the dispersion-relation domain, since triplications are impossible there. Figure 3 shows how the three approximations shown in Figure 2 work in the dispersion-relation domain. Just like elliptical anisotropy, Muir's four-parameter approximation is calculated in the dispersion-relation domain by simply using slownesses instead of velocities in the same approximation formula. Note in Figure 3 that the double-elliptic approximation in the dispersion-relation domain is nearly exact for this example.