The double-elliptic approximation in the group and phase domains

Conclusions

The double-elliptic approximation introduced by Muir (1990) can approximate the kinematics of TI media quite accurately, although it is forced to cut off triplications when fit in the impulse-response domain. In the dispersion-relation domain it remains accurate even for triplicating media.

Our analysis in this paper was limited to simple kinematics. Before pronouncing the double-elliptic approximation a success, we need to also demonstrate how it works when used as the basis for an imaging technique; i.e, how accurately does it model the dynamics of the wave equation? Karrenbach (1991) examines this question in a companion paper.

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Figure 4. Two different double-elliptic approximations (dashed curves) fit to the qSV mode of Greenhorn Shale (solid curves). Left: the approximation is fit in the impulse-response domain, and so the dashed curve has a simple analytic form. Right: the approximation is fit in the dispersion-relation domain, and so is able to closely follow the triplication. This approximating curve can only be calculated parametrically, however, and so is less useful.

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Figure 5. Two different double-elliptic approximations (dashed curves) corresponding to those in Figure 4, but this time fit to the qP mode of Greenhorn Shale (solid curves). (The size of the ``$*$'' in the middle shows the relative scales.) The discrepancy is much less since there are no troublesome triplications.

The double-elliptic approximation in the group and phase domains