Equations 1 and 4 serve as the basis for our construction in this study, where we seek the following second-order total one-way traveltime derivative,
evaluated at with respect to the reference vertical ray in a 1-D anisotropic medium (Figure 1). We denote the reference intersection point by and use `capital letters' to denote quantities evaluated with respect to the reference 1-D anisotropic medium.In consideration of reflection traveltime, equation 5 is related to the NMO velocity as follows:
where is the one-way zero-offset traveltime from the point on the reflector to the surface. In the case of diffraction traveltime, equation 5 can be related to time-migration velocity as follows: where is the one-way vertical traveltime from the point scatterer to the surface and we replace with . Even though and have the same mathematical expression, they represent two different quantities. In case of reflection traveltime, it is common to use to denote half-offset. On the other hand, in consideration of diffraction traveltime, we use here to denote the distance between the escape location of image ray and any surrounding point on the surface. We give a brief review on the derivation of equations 6 and 7 in Appendix A.Because both NMO velocity (equation 6) for reflection traveltime and time-migration velocity (equation 7) for diffraction traveltime are related to the one-way traveltime derivative of a ray (either normal-incidence or image ray), it is, in principle, sufficient to study the effects from lateral heterogeneity on such derivative of a generic ray travelling between a subsurface position and the surface. We shall adopt this notion in this paper and begin our study by first looking at how the desired second-order one-way traveltime derivative evaluated at the surface (equation 5) is related to the derivatives of in each sublayer based on the Fermat's principle. This relationship will be used to accumulate the total effects of lateral heterogeneity from all sublayers.