# Using an upward-traveling coordinate system

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 366 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 9.28

Show that a change from one-way traveltime $t$ to a new variable $t*$ ,

 {\begin{aligned}t^{*}=t-z/V,\end{aligned}} (9.28a)

changes from a space-fixed coordinate system $(x,\;z,\;t)$ to a coordinate system $(x,\;z,\;t^{*})$ that “rides along on an upcoming wavefront.” This transform is used in migration. Assume that the velocity $V$ is constant and $z$ is the depth.

### Solution

Consider a wavefront that passes through the point $(x_{0},\;z_{0})$ at time $t=t_{0}$ ; at this instant, $t_{0}^{*}=t_{0}-z_{0}/V$ . After a further interval $\Delta t$ , the wavefront has advanced upward the distance $\Delta z=V\Delta t$ , so

{\begin{aligned}t^{*}=(t-\Delta t)-(z_{0}-\Delta z)/V=(t_{0}-z_{0}/V)-(\Delta t-\Delta z/V).\end{aligned}} Since $\Delta z=V\Delta t$ , $t^{*}=t_{0}^{*}$ , that is, the coordinate system keeps pace with the upcoming wavefront.