# Delay time

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Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 11 415 - 468 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 11.8

Show that ${\displaystyle NQ}$ in Figure 11.8a is given by

 {\displaystyle {\begin{aligned}NQ=V_{2}\delta _{NQ}\tan ^{2}\theta _{c},\\{\hbox{i.e.,}}\quad \quad \delta _{g}=\delta _{NQ}=NQ/V_{2}\tan ^{2}\theta _{c}.\end{aligned}}} (11.8a)

### Background

The concept of delay time has found wide application in refraction interpretation (see problems 11.9 and 11.11). We define the delay time associated with the refraction path SMNG in Figure 11.8a as the observed traveltime minus the time required to travel from ${\displaystyle P}$ to ${\displaystyle Q}$ at the velocity ${\displaystyle V_{2}}$. ${\displaystyle PQ}$ is the projection of the path SMNG onto the refractor), Writing ${\displaystyle \delta }$ for the total delay time, we have

 {\displaystyle {\begin{aligned}\delta &=t_{SG}-PQ/V_{2}\end{aligned}}} (11.8b)

{\displaystyle {\begin{aligned}&=\left({\frac {SM+NG}{V_{1}}}+{\frac {MN}{V_{2}}}\right)-{\frac {PQ}{V_{2}}}=\left({\frac {SM}{V_{1}}}-{\frac {PM}{V_{2}}}\right)+\left({\frac {NG}{V_{1}}}-{\frac {NQ}{V_{2}}}\right)\end{aligned}}}

 {\displaystyle {\begin{aligned}&=\delta _{s}+\delta _{g},\end{aligned}}} (11.8c)

 {\displaystyle {\begin{aligned}{\hbox{where}}\quad \quad \delta _{s}={\hbox{source delay time}}\ =\left({\frac {SM}{V_{1}}}-{\frac {PM}{V_{2}}}\right)\end{aligned}}} (11.8d)

 {\displaystyle {\begin{aligned}{\hbox{and}}\quad \quad \delta _{g}={\hbox{geophone delay time}}\ =\left({\frac {NG}{V_{1}}}-{\frac {NQ}{V_{2}}}\right).\end{aligned}}} (11.8e)

### Solution

Referring to Figure 11.8a, we have, by definition,

{\displaystyle {\begin{aligned}\delta _{g}&=NG/V_{1}-NQ/V_{2}\\&=NQ\left({\frac {1}{V_{1}\sin \theta _{c}}}-{\frac {1}{V_{2}}}\right)={\frac {NQ}{V_{2}}}\left({\frac {1}{\sin ^{2}\theta _{c}}}-1\right)\\&=NQ/V_{2}\tan ^{2}\theta _{c}.\end{aligned}}}

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