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Introducing the offset continuation equation

Most of the contents of this paper refer to the following linear partial differential equation:
\begin{displaymath}
h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2 ...
...\, {\partial^2 P \over {\partial t_n \,
\partial h}} \,\,\, .
\end{displaymath} (1)

Equation (1) describes an artificial (non-physical) process of transforming reflection seismic data $P(y,h,t_n)$ in the offset-midpoint-time domain. In equation (1), $h$ stands for the half-offset ($h=(r-s)/2$, where $s$ and $r$ are the source and the receiver surface coordinates), $y$ is the midpoint ($y=(r+s)/2$), and $t_n$ is the time coordinate after normal moveout correction is applied:
\begin{displaymath}
t_n=\sqrt{t^2-{4 \, h^2 \over v^2}}\;.
\end{displaymath} (2)

The velocity $v$ is assumed to be known a priori. Equation (1) 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.



Subsections
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Next: Proof of validity Up: Theory of differential offset Previous: Introduction

2014-03-26