Introducing the offset continuation equation

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

$$h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2 ... ...n \, {\partial^2 P \over {\partial t_n \, \partial h}} \,\,\, .$$ (140)

Equation () describes an artificial (non-physical) process of transforming reflection seismic data P(y,h,t_n) in the offset-midpoint-time domain. In equation (), 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:  

$$\left(t_n=\sqrt{t^2-{4 \, h^2 \over v^2}}\right)\;.$$ (141)

The velocity v is assumed to be known a priori. Equation () belongs to the class of linear hyperbolic equations, with the offset h acting as a time-like variable. It describes a wave-like propagation in the offset direction.