Point of maximum depth

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
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 with the positive y-axis (Figure 13). This raypath represents a diving wave that reaches a maximum depth at point and then returns to the surface. The tangent to the raypath is horizontal at D. Thus, the angle from the vertical is , that is, at the depth point. Inserting and in the above parametric equations, we obtain the coordinates of the depth point as


(56)

Let denote the traveltime along the raypath from the origin to the point of maximum depth. At the point of maximum depth, the angle is . Thus, the implicit equation for the traveltime is


(57)
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


(58)

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 . Similarly, the hyperbolic sine is defined as . We thus have


(59)

and, similarly,


(60)

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


(61)


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