Delay time

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

Problem 11.8

Show that $$ NQ $$ in Figure 11.8a is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

(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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q at the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{2} . $$ PQ $$ is the projection of the path SMNG onto the refractor), Writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \delta for the total delay time, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \delta &=t_{SG} -PQ/V_{2} \end{align}

(11.8b)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &=\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{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} &=\delta_{s} + \delta_{g}, \end{align}

(11.8c)

{\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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{and}\quad \quad \delta_{g} = \hbox{geophone delay time}\ = \left(\frac{NG}{V_{1} } -\frac{NQ}{V_{2} } \right). \end{align}

(11.8e)

Solution

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \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{align}

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Proof of a generalized reciprocal method relation	Barry’s delay-time refraction interpretation method
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Geologic interpretation of reflection data	3D methods
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