@Flojo1 - I know more about RF atmospheric refraction than optical, so this won't give you anywhere near a complete answer. But I can say a few things that may be of some use.
First, looming is no more than enough vertical refraction in the earth atmosphere that an object on the surface looks like it is above the surface. Sinking is the opposite. There's other refraction effects that also include distortion. Normal refraction (refraction in a standard atmosphere) is that light rays bend downward and thus are able to propagate past the geometrical horizon as the earth's surface curves down. A basic description is in http://www.smithsonianmag.com/science-nature/did-the-titanic-sink-because-of-an-optical-illusion-102040309/ and actually includes some numbers on propagation for astronomical observations for angles 0 to 20 degrees off the horizon, and so more than 10 Kms.
The refractivity N (a millionth of n-1, with n the index of refraction) is used for atmospheric refraction where n is not too far from 1. N rates of change with altitude, the lapse rate, of about -40 units per km is a normal atmosphere, where n goes lower with altitude. The air density also decreases with altitude. This for a standard atmosphere. For N rates at -157 units/km the wave would follow the earth curvature and theoretically go all the way around. In between it is super refractive. The last reference below shows this. Other references are below. Of course n changes with density and with air composition, and with wavelength. Density changes with p and T.
Fact is that this has been getting measured and modeled for over 100 years. I've seen more RF models than optical ones but you just need to do more looking/research. Some of it, for optical, may be related to laser and other atmospheric propagation for defense purposes, so not as publicized.
Of course lots of studies also on the variation of n with various different concentrations of other chemicals in the atmosphere, though normal air and water vapor are the biggest factors.
http://www-rohan.sdsu.edu/~aty/mirages/mirsims/loom/loom.html Shows looming effects
https://en.m.wikipedia.org/wiki/Atmospheric_refraction. Shows atmospheric refraction and has some equations, including for views long views at angles 0 to 20 degrees above the horizon
http://www.mike-willis.com/Tutorial/PF6.htm Has super refraction