Tiempo de viaje para velocidad lineal con la profundidad

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This page is a translated version of the page Traveltime for velocity linear with depth and the translation is 100% complete.
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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 2
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

La traveltime es


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} t&{\rm =}\int^{\theta {\rm /}pa}_{{\theta }_0/pa}{\frac{ds}{v}} {\rm =}\frac{{\rm 1}}{a}\int^{\theta }_{{\theta }_0}{\frac{d\theta }{{\rm \ sin\ }\theta }}. \end{align} (52)

La integración proporciona el traveltime como


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} &t{\rm =}{\left[\frac{{\rm l}} {a}{\rm \ log\ }\frac{{\rm 1}-{\rm \ cos\ }\theta }{{\rm \ sin\ }\theta }\right]}^{\theta }_{{\theta }_0}{\rm =}{\left[\frac{{\rm l}}{a}{\rm \ log\ \ tan\ }\frac{\theta }{{\rm 2}}\right]}^{\theta }_{{\theta }_0}{\rm =}\frac{{\rm l}}{a}{\rm \ log\ }\frac{{\rm \ tan\ }\left(\theta {\rm /2}\right)}{{\rm \ tan\ }\left({\theta }_0{\rm /2}\right)}, \end{align} (53)

lo cual da


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} at&{\rm =\ log\ }\frac{{\rm \ tan\ }\left(\theta {\rm /2}\right)}{{\rm \ tan\ }\left({\theta }_0{\rm /2}\right)} \end{align} (54)

Si tomamos la exponencial de ambos lados, obtenemos la ecuación implícita para traveltime dada por


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} e^{at}&{\rm =}\frac{{\rm \ tan\ }\left(\theta {\rm /2}\right)}{{\rm \ tan\ }\left({\theta }_0{\rm /2}\right)}. \end{align} (55)


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Traveltime for velocity linear with depth/es