To simplify the work, the object should have a simple shape. I've chosen a triangle with a 90° vertex at the top, so that the sides slope at 45°. In addition, I've painted some stripes on the triangle parallel to its left edge; these are sometimes useful in distinguishing inverted and erect portions of the miraged images. Although one occasionally sees conical hills with a triangular profile, they don't have stripes; so I'll call my object a “target”, to emphasize its artificiality.
To understand mirages, it's very useful to know what the scene looks like without a mirage. So here's a simulation of the target's appearance as seen from a height of 1.5 m through just 2 km of the Standard Atmosphere. This is essentially an undistorted view of the target.
The vertical scale, at the left side, is height in the atmosphere. You can read off the observer's height by noticing where on this scale the rays all converge. The horizontal scale is horizontal distance; the plot is correct in rectangular coordinates, so the heavy line that represents the Earth's surface is concave downward. Obviously, the ray diagram is grossly exaggerated in the vertical direction, to make its details visible. (Note that the vertical scale is in meters, and the horizontal one, in km.) The target here has twice the height of the observer's eye; this puts the middle of the target at eye level.
Notice that the vertical scales on the transfer curve and the ray diagram are identical. However, the target is at the right side of the ray diagram; as the vertical scale of the transfer curve has its zero level at the surface of the Earth, which is lower in the ray diagram at the target than at the observer, this is offset from the vertical coordinate of the ray diagram.
In the example shown, it's clear that the target is much closer than the sea horizon, which is seen in the distance behind it. The target resembles a triangular island rising from the sea. The overall effect is not very realistic, but is suggestive enough to give an impression of what the atmosphere might do (in this case, nothing) to the appearance of a simple object.
In addition, there are simulations of looming, towering, sinking and stooping, although these related refraction phenomena are not themselves mirages.
Notice that the size of the target changes from one set of simulations to the next. For the inferior-mirage simulations, the observer is close to the surface; the apparent horizon is not far away; and a small target, only 3 m high, is used, to make the mirage effects plain. (The same small target is used in the Standard-Atmosphere and looming simulations.) But for the mirages that involve ducts, like the mock mirage and superior-mirage simulations, a much larger target height (200 m) is appropriate.
The relation between the angular altitude of each ray at the eye to its height at the target is the transfer curve for the mirage. Given the transfer curve, it's easy to draw the apparent distorted shape of the target seen through the refracting atmosphere.
As with the sunset simulations, I have a program that generates the output drawing as a PostScript figure. I then convert that to a PNG file for the appropriate Web page.
Copyright © 2008 – 2012 Andrew T. Young
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