INTRODUCING THE OFFSET CONTINUATION EQUATION

Most of the contents of this paper refer to the following linear partial differential equation:  

$$h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2 ... ...n \, {\partial^2 P \over {\partial t_n \, \partial h}} \,\,\, .$$ (1)

Equation (1) describes an imaginary (nonphysical) process of reflection seismic data transformation in the offset-midpoint-time domain. Here h stands for the half-offset (h=(r-s)/2, where s and r are the source and the receiver coordinates), y is the midpoint (y=(r+s)/2), and t_n is the time coordinate after normal moveout correction is applied.

Equation (1) and the previously published OC equation Bolondi et al. (1982) differ only with respect to the single term $\partial^2 P \over {\partial h^2}$. However, this difference is substantial. As Appendix A proves, the range of validity for the approximate OC equation  

$$h \, {\partial^2 P \over \partial y^2} \, = \, t_n \, {\partial^2 P \over {\partial t_n \, \partial h}}$$ (2)

can be defined by the inequality  

$$h/z \ll \cot{\alpha} \, ,$$ (3)

where z is the reflector depth, and $\alpha$ is the dip angle. For example, for a dip of 45 degrees, equation (2) is valid only for offsets that are much smaller than the depth.

In order to prove the theoretical validity of equation (1) for all offsets and reflector dips, I apply a simplified version of the ray method technique Babich (1991); Cerveny et al. (1977) and obtain two equations to describe separately wavefront (traveltime) and amplitude transformation in the OC process. 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  

$$P\left(y,h,t_n\right) \approx A_n(y,h)\,R_n\left(t_n-\tau_n(y,h)\right) \;,$$ (4)

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 (4) as a trial solution for (1), collecting terms that have the same asymptotic order, and neglecting low-order terms produces a set of two first-order partial differential equations:  

$$h \, \left[ {\left( \partial \tau_n \over \partial y \right)... ... = \, - \, \tau_n \, {\partial \tau_n \over \partial h} \,\,\,,$$ (5)

 

$$\left( \tau_n - 2h \, {\partial \tau_n \over {\partial h}} \... ...rtial^2 \tau_n \over {\partial h^2}} \right) \, = \, 0 \,\,\,.$$ (6)

Equation (5) describes the transformation of traveltime curve geometry in the OC process analogously to the eikonal equation in the wavefront propagation theory. Thus, what appear to be wavefronts of the wave motion described by (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 (6). In terms of the theory of partial differential equations, equation (5) is the characteristic equation for (1).