I'm a geoscientist and am trying to figure out how the slopes of the flanks of a hyperboloid change for straight lines that cross them.
The line of reference is through the apex (red). Now take any of the other parallel lines that do not cross the apex. Is the slope of their flanks the same as for the line through the apex?
Or is it steeper?
My gut-reaction is that the slope is the same
Any insight from experts would be helpfull; thanks.
The backstory:
In seismic processing you can predict the wavefield of a recorded multiple by convolving two already recorded wavefield of primary reflections.
See the following picture:
The resulting wavefield for the 2D case is a hyperbole.
By stacking it you get the multiple prediction for 1 source-receiver-pair.
However; if the subsurface is not 1 or 2 dimensional, but 3D then the predicted multiple-wavefield is a hyperboloid.
So if you only measured on this one 2D-line instead of doing a whole 3D survey then your predicted multiple-wavefield would be a hyperbole that does not go through the apex of the hyperboloid; basically the green line instead of the red line.
Stacking it would give you the wrong arrival time for the multiple-trace, and also due to different reflection-angles in the subsurface a wrong amplitude. However, casting these errors aside, I am interested about the slope of the different hyperbolas; steeper slopes would result in a weaker multiple which would be an additional error.
Thanks for you patience.
ONCE AGAIN: I CANNOT UPLOAD THE PICTURES; SEE THE FOLLOWING LINK FOR A FOLDER CONTAINING THEM ADDITIONAL PICTURES
The hyperboloid as pictured has equation of the form $$ z = -\sqrt{a^{2} + x^{2} + y^{2}} $$ for some real $a > 0$.
The red curve through the apex is the slice $y = 0$, whose equation and derivative (slope) as functions of $x$ are $$ z = -\sqrt{a^{2} + x^{2}},\qquad \frac{dz}{dx} = -\frac{2x}{\sqrt{a^{2} + x^{2}}}. \tag{1} $$
The green curve is the slice $y = c$, whose equation and derivative (slope) as functions of $x$ are $$ z = -\sqrt{(a^{2} + c^{2}) + x^{2}},\qquad \frac{dz}{dx} = -\frac{2x}{\sqrt{(a^{2} + c^{2}) + x^{2}}}. \tag{2} $$
These curves are both hyperbolas, but they are not congruent, so they do not have the same slope. (Without the square root, the sections would be congruent parabolas, in case that matters.)
To confirm this claim algebraically, translate (2) vertically by $\sqrt{a^{2} + c^{2}} - a$, and note that $$ z = -\sqrt{a^{2} + x^{2}},\qquad z = \sqrt{a^{2} + c^{2}} - a -\sqrt{(a^{2} + c^{2}) + x^{2}} $$ agree at $x = 0$, but are not the same function. (For example, the first has $z = \pm x$ as asymptotes, while the second has asymptotes $z = \sqrt{a^{2} + c^{2}} - a \pm x$.)
The graph below shows several members of a typical family of slices, translated to a common maximum height. The larger the value of $c$ (i.e., the farther the slice is taken from $y = 0$), the "flatter" the hyperbola.