# 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 ${\displaystyle t}$ to a new variable ${\displaystyle t*}$,

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

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

### Solution

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

{\displaystyle {\begin{aligned}t^{*}=(t-\Delta t)-(z_{0}-\Delta z)/V=(t_{0}-z_{0}/V)-(\Delta t-\Delta z/V).\end{aligned}}}

Since ${\displaystyle \Delta z=V\Delta t}$, ${\displaystyle t^{*}=t_{0}^{*}}$, that is, the coordinate system keeps pace with the upcoming wavefront.