conclusions

We have proposed a general framework for evaluating the one-way traveltime derivatives in layered anisotropic media in the presence of weak lateral heterogeneity from curved reflectors and lateral velocity variations. Relying on the Fermat's principle, we show that in 2D media, the effects from lateral heterogeneity in each sublayer gets accumulated according to a recursive relationship. We specify the expressions for the second-order traveltime derivatives related to NMO velocity and time-migration velocity suitable for 2D datadets that honor these effects from lateral heterogeneity. We subsequently show that the new expressions can lead to improved moveout predictions and better focused diffraction responses. The results of this study fill in the missing gap in existing theory and serve as a basis for understanding and quantifying the first-order effects of lateral heterogeneity on reflection and diffraction traveltimes in multi-layer anisotropic media. The fundamentals provided in this work is important for future design of accurate parameter estimation techniques such as Dix-type inversion that honor the effects from lateral heterogeneity.