I figured out a good approximation formula to calculate R if we know the distance to an object (D), how much of the object is observed to be hidden (x) and the observer height (h) without having to know the distance to the horizon. This values can be figured out by researching the target without even having to go there or to the horizon. My formula is:
This formula can now be used for simple observations. D, x, and h are easy measurable without having to go to the horizon or target, if we know the size and elevation of the target. The factor 6/7 is the correction for standard refraction.
Example: instead of real measurement I use some values from my Curve Calculator to check the validity of this equation. I set refraction as standard 7/6 R.
x would be the observed hidden height from known target size minus the visible part.
Plug this in into the equation above gives:
This is the apparent radius of the earth due to refraction. Applying the refraction correction 6/7 gives:
Here is a practical example demonstrated: www.youtube.com. See my pinned comment where I calculated R = 6398 km.
Here is how I derived this formula:
1) Using Pythagoras we get an equation for the distance to the horizon d:
2) Because in most cases 2Rh is much, much bigger than h2, h2 is negligible and we get the very good approximation:
3) We now have a formula for the relation of distance to horizon d1 and height h. The same formula can be used for the relation of the distance of the target to the horizon d2 and the hidden height x:
Now the known distance from the observer to the target is D = d1 + d2, so we can add the 2 equations and get:
4) Here we have all values known except R, which we want to figure out. This equation is easy to solve for R:
Now add the refraction correction factor 6/7 to correct for refraction.