Theory of differential offset continuation

Proof of validity

A simplified version of the ray method technique (Babich, 1991; Cervený, 2001) can allow us to prove the theoretical validity of equation (1) for all offsets and reflector dips by deriving two equations that describe separately wavefront (traveltime) and amplitude transformation. According to the formal ray theory, the leading term of the high-frequency asymptotics for a reflected wave recorded on a seismogram takes the form

$$P\left(y,h,t_n\right) \approx A_n(y,h)\,R_n\left(t_n-\tau_n(y,h)\right) \;,$$ (3)

where $A_n$ stands for the amplitude, $R_n$ is the wavelet shape of the leading high-frequency term, and $\tau_n$ is the traveltime curve after normal moveout. Inserting (3) as a trial solution for (1), collecting terms that have the same asymptotic order (correspond to the same-order derivatives of the wavelet $R_n$), and neglecting low-order terms, we arrive at the set of two first-order partial differential equations:

$$h \, \left[ {\left( \partial \tau_n \over \partial y \right)... ... \, - \, \tau_n \, {\partial \tau_n \over \partial h} \,\,\,,$$ (4)

$$\left( \tau_n - 2h \, {\partial \tau_n \over {\partial h}} \... ...tial^2 \tau_n \over {\partial h^2}} \right) \, = \, 0 \,\,\,.$$ (5)

Equation (4) describes the transformation of traveltime curve geometry in the OC process analogously to how the eikonal equation describes the front propagation in the classic wave theory. What appear to be wavefronts of the wave motion described by equation (1) are traveltime curves of reflected waves recorded on seismic sections. The law of amplitude transformation for high-frequency wave components related to those wavefronts is given by equation (5). In terms of the theory of partial differential equations, equation (4) is the characteristic equation for (1).

Theory of differential offset continuation