Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

Reflection traveltime in any type of model can be considered as a
function of the source and receiver locations and and the
location of the reflection point , as follows:

where is the traveltime of the direct wave. Clearly, at the zero-offset point,

where corresponds to the reflection point of the zero-offset ray.

Differentiating equation (A-1) with respect to the half-offset
and applying the chain rule, we obtain

Equation (A-5) must be satisfied for any values of and . Substituting this equation into equation (A-4) leads to the equation

Differentiating (A-6) again with respect to , we arrive at the
equation

Substituting the expression for the function (A-2) into (A-8) leads to the equation

which is the mathematical formulation of the NIP theorem. It proves that the second-order derivative of the reflection traveltime with respect to the offset is equal, at zero offset, to the second derivative of the direct wave traveltime for the wave propagating from the incidence point of the zero-offset ray. One immediate conclusion from the NIP theorem is that the short-spread normal moveout velocity, connected with the derivative in the left-hand-side of equation (A-9) can depend on the reflector dip but doesn't depend on the curvature of the reflector. Our derivation up to this point has followed the derivation suggested by Chernjak and Gritsenko (1979).

Differentiating equation (A-7) twice with respect to evaluates,
with the help of the chain rule, the fourth-order derivative, as
follows:

Again, we can apply the principle of reciprocity to eliminate the odd-order derivatives of in equation (A-11) at the zero offset. The resultant expression has the form

In order to determine the unknown second derivative of the reflection point location , we differentiate Fermat's equation (A-5) twice, obtaining

Simplifying this equation at zero offset, we can solve it for the second derivative of . The solution has the form

Here we neglect the case of , which corresponds to a focusing of the reflected rays at the surface. Finally, substituting expression (A-14) into (A-12) and recalling the definition of the function from (A-2), we obtain the equation

which is the same as equation (67) in the main text. Higher-order derivatives can be expressed in an analogous way with a set of recursive algebraic functions (Fomel, 1994).

In the derivation of equations (A-9) and (A-15), we have used Fermat's principle, the principle of reciprocity, and the rules of calculus. Both these equations remain valid in anisotropic media as well as in heterogeneous media, providing that the traveltime function is smooth and that focusing of the reflected rays doesn't occur at the surface of observation.

Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

2014-01-27