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Discussion

In this study, we have focused only on pure-mode reflections where the source-receiver reciprocity holds and the moveout approximation around zero-offset is an even function (equations 3 and 4). This symmetry is generally absent in the case of converted waves.

In addition, we assume that there is no relection dispersal, which is equivalent to assuming that the one-way traveltime can be expressed in terms of half-offset only $ t(h)$ . This assumption is appropriate when the two legs of the ray path are symmetric with respect to zero-offset and is related to the case of laterally homogeneous, horizontal anisotropic layers with a horizontal symmetry plane. When this assumption does not hold, for example, where tilted anisotropic media without horizontal symmetry planes or arbitrarily shaped interfaces are considered, one-way traveltime is no longer a function of merely half-offset $ h$ but also a function of reflection point $ y(h)$ . Grechka and Tsvankin (1998) emphasized that the effect of reflection dispersal can be neglected in consideration of NMO velocity (Hubral and Krey, 1980). However, reflection dispersal becomes important when higher-order coefficients are considered. The general form for the quartic and higher-order coefficients that honor reflection dispersal effects was first studied by Fomel (1994) based on the extension of Normal-Incident-Point (NIP) theorem (Gritsenko, 1984) to the higher-order. Similar derivation for the quartic coefficientis given by Pech et al. (2003).

In general, with the assumption of laterally homogeneous and horizontal layers, one can conveniently express one-way traveltime and half-offset in each sublayer as functions of horizontal slownesses ($ p_i$ ) as shown in equations 9 and 10. Their derivatives in the most general form are given in equations 17-20. In consideration of pure modes, the zero-offset corresponds to $ p_i=0$ and, which results in the simplified expressions in equations 21-24. In the case of converted waves or lateral variations or anisotropic media without horzontal symmetry plane, the location of $ p_i=0$ no longer corresponds to the zero-offset but instead to the location of the minimum traveltime. The traveltime derivatives in equations 17-20 can still be used with proper specifications of which position for the derivatives to be evaluated at. On the other hand, if the interfaces are arbitrarily shaped, the appropriate traveltime derivatives can be computing from ray tracing with the knowledge of approximate group velocity in each sublayer (Sripanich and Fomel, 2014,2015b). The proposed framework has a straightforward extension from quartic to higher-order coefficients as long as all the mentioned assumptions are satisfied.


next up previous [pdf]

Next: Conclusions Up: Sripanich & Fomel: Interval Previous: Coefficients of traveltime expansion

2017-04-14