Point of maximum depth

**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

We have found the parametric equations for the circular raypath that starts at the origin and makes an angle 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/":): {\theta }_0 with the positive y-axis (Figure 13). This raypath represents a diving wave that reaches a maximum depth at point 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/":): D{\rm =}\left(x_D{\rm ,\ }y_D\right) and then returns to the surface. The tangent to the raypath is horizontal at D. Thus, the angle from the vertical is 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/":): {\rm 9}0^{{\rm o}} , that is, 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/":): \theta {\rm =} {\theta }_D{\rm =} \pi {\rm /2} at the depth point. Inserting 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/":): {\rm \ cos\ }\ \theta {\ =\ cos\ }{\theta }_D{\ =\ 0} and 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/":): {\rm \ sin\ }\theta {\ =\ sin\ }{\theta }_D{\ =} {\ 1} in the above parametric equations, we obtain the coordinates of the depth point as

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} x_D&{\rm =}\frac{v_0}{a{\rm \ tan\ }{\theta }_0} \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} y_D&{\rm =}\frac{v_0}{a{\rm \ sin\ }{\theta }_0}-\frac{v_0}{a}{\rm =}\frac{v_0}{a}\left(\frac{{\rm l}} {{\rm \ sin\ }{\theta }_0}-{\rm 1}\right) . \end{align}

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Let 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/":): t_D denote the traveltime along the raypath from the origin to the point of maximum depth. At the point of maximum depth, the angle is 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/":): \theta {\rm =} {\theta }_D{\rm =}\pi {\rm /2} . Thus, the implicit equation for the traveltime 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/":): t_D is

{\begin{aligned}e^{at_{D}}&{\rm {=}}{\frac {{\rm {\ tan\ }}{\frac {{\theta }_{D}}{\rm {2}}}}{{\rm {\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}}}{\rm {=}}{\frac {{\rm {\ tan\ }}{\frac {\pi }{\rm {4}}}}{{\rm {\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}}}{\rm {=}}{\frac {\rm {l}}{{\rm {\ tan\ }}{\frac {{\theta }_{0}}{\rm {2}}}}}{\rm {=}}{\frac {\rm {l}}{\frac {{\rm {1}}-{\rm {\ cos\ }}{\theta }_{0}}{{\rm {\ sin\ }}{\theta }_{0}}}}{\rm {=}}{\frac {{\rm {\ sin\ }}{\theta }_{0}}{{\rm {l}}-{\rm {\ cos\ }}{\theta }_{0}}},\end{aligned}}

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**Figure 13.** A diving wave from origin O to maximum depth D and back to the horizontal axis. It is called a diving wave because it plunges into the ground and comes up to the surface, not by reflection but by making a U-turn. Because depth is plotted upward, the diving wave appears inverted. The wavefront though the maximum depth point is also shown.

which also can be written as

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_D}&{\rm =\ tan\ }\frac{{\theta }_0}{{\rm 2}} {\rm =}\frac{{\rm l}-{\rm \ cos\ }{\theta }_0}{{\rm \ sin\ }{\theta }_0}. \end{align}

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These implicit equations involve exponentials. There are two common ways to get rid of exponentials: by logarithms and by hyperbolic functions. We choose the latter. The hyperbolic cosine is defined as ${\rm {\ cosh\ }}u{\rm {=}}\left(e^{\mathbf {u} }{\rm {+}}e^{-\iota \iota }\right){\rm {/2}}$ . Similarly, the hyperbolic sine is defined as 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/":): {\rm \ sinh\ }u{\rm =}\left(e^{\mathbf{u}}-e^{-\mathbf{u}}\right){\rm /2} . We thus 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} {\rm \ cosh\ }at_D{\rm =}\frac{e^{at_D}{\rm +}e^{-at_D}} {{\rm 2}}{\rm =}\frac{\frac{{\rm \ sin\ }{\theta}_0}{{\rm 1}-{\rm \ cos\ }{\theta }_0}{\rm +}\frac{{\rm 1}-{\rm \ cos\ }{\theta }_0}{{\rm \ sin\ }{\theta }_0}}{{\rm 2}}{\rm =}\frac{{{\rm sin}}^{{\rm 2}}{\theta }_0{\rm +}{\left({\rm 1}-{\rm \ cos\ }{\theta }_0\right)}^{{\rm 2}}}{{\rm 2}\left({\rm l}-{\rm \ cos\ }{\theta }_0\right){\rm \ sin\ }{\theta }_0}{\rm =}\frac{{\rm l}}{{\rm \ sin\ }{\theta }_0} \end{align}

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and, similarly,

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} {\rm \ sinh\ }at_D&{\rm =}\frac{e^{at_D}-e^{-a{{\rm t}} _O}}{{\rm 2}}{\rm =}\frac{{\rm l}}{{\rm 2}}\left(\frac{{\rm \ sin\ }{\theta }_0}{{\rm 1}-{\rm \ cos\ }{\theta }_0}-\frac{{\rm 1}-{\rm \ cos\ }{\theta }_0}{{\rm \ sin\ }{\theta }_0}\right){\rm =}\frac{{\rm l}}{{\rm \ tan\ }{\theta }_0}. \end{align}

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In terms of traveltime to the point of maximum depth, the point of maximum depth is given by the coordinates

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} x_D{\rm =}\frac{v_0}{a{\rm \ tan\ }{\theta }_0}{\rm =}\frac{v_0}{a}{\rm \ sinh\ }at_D \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} y_D&{\rm =}\frac{v_0}{a{\rm \ sin\ }{\theta }_0}-\frac{v_0}{a}{\rm =}\frac{v_0}{a}\left({\rm \ cosh\ }at_D-{\rm 1}\right) . \end{align}

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