    Non-hyperbolic common reflection surface  Next: Appendix B: Circular reflector Up: Fomel & Kazinnik: Nonhyperbolic Previous: Acknowledgments

# Apendix A: Hyperbolic reflector

In this appendix, we reproduce the derivation of an analytical expression for reflection traveltime from a hyperbolic reflector in a homogeneous velocity model (Fomel and Stovas, 2010). Similar derivations apply to an elliptic reflector and were used previously in the theory of offset continuation (Stovas and Fomel, 1996; Fomel, 2003).

Consider the source point and the receiver point at the surface above a 2-D constant-velocity medium and a hyperbolic reflector defined by the equation (19)

The reflection traveltime as a function of the reflection point location is (20)

According to Fermat's principle, the traveltime should be stationary with respect to the reflection point :     (21)

Putting two terms in equation (A-3) on the different sides of the equation, squaring them, and reducing their difference to a common denominator, we arrive at the following quadratic equation with respect to :     (22)

Only one of the two branches of the solution has physical meaning. Substituting this solution into equation (A-2), we obtain, after a number of algebraic simplifications,     (23)

Making the variable change in equation (A-5) from and to midpoint and half-offset coordinates and according to , , we transform this equation to form (12), where the following correspondence between parameters is applied:   (24)   (25)   (26)   (27)

The inverse relationships are given by   (28)   (29)   (30)   (31)

The connection with the multifocusing parameters is summarized in Table 1 for the general case and three special cases (a plane dipping reflector, a flat reflector, and a point diffractor). The first two special cases turn the nonhyperbolic CRS equation into the hyperbolic form (8). The last case turns it into the double-square-root form (13).    Hyperbolic reflector    Plane dipping reflector   0 Flat reflector   0 0 Point diffractor     Table 1. Multifocusing parameters for a hyperbolic reflector in a homogeneous medium ( ). The general case and three special cases.    Non-hyperbolic common reflection surface  Next: Appendix B: Circular reflector Up: Fomel & Kazinnik: Nonhyperbolic Previous: Acknowledgments

2013-07-26