Looking for more derivations of the 7/6 (1.1666) value.

https://en.wikipedia.org/wiki/Atmospheric_refraction

The equation cited is from Hirt, et al, 2010, and appears to have been empirically derived from a long series of simultaneous reciprocal observations and back-calculating the refraction component from those observations. They then built an empirical relationship between the temperature gradient and the refraction correction. I haven't read their paper, but the technique of calculating refraction corrections from simultaneous reciprocal observation appears to have the earth's radius as one of its assumptions, so FE doing a deep dive into it will find a somewhat circular reasoning.

One reason you are getting much higher

*k* value than is normally used is you plugged in a temperature inversion when you used a positive value for

*dT/dh*. In "standard conditions" the temperature drops as you get higher, so

*dT/dh* is negative. Using

*dT/dh* = -0.0065 gets to within about 3% of the 1/6 of curvature drop rule of thumb for standard refraction. If you use a very strong negative gradient, the value of

*k* goes negative, which is what you see with a road "mirroring" the sky on a hot sunny day.

A derivation from basic principles of a similar refraction correction equation (with an added humidity gradient correction) is found in Torge's "Geodesy". In the 2nd edition (1991) the derivation is in section 4.3.1, and the entire section is in Google Books' preview of the book. In the 3rd edition it is section 5.1, 5.1.1, and 5.1.2, but in Google Books' preview two pages of these sections are omitted, so you will have to look for those pages in a library. The two derivations differ slightly in presentation, and the third edition relies on a new set of atmospheric index of refraction equations adopted by the International Association of Geodesy in 1999, but the end result is the same, as several of the coefficients have been rounded and the newer index of refraction model was in general agreement with the previously used equations. Torge's derivation ends with equation 4.40 (2nd ed) or 5.19a (3rd ed), which doesn't quite give it in a form where

*k* is calculated directly. Baselga 2014 takes that and plugs it into one of the preceding equations in Torge's derivation to present

Baselga et al said:

*k* = (R / 10^6) * {78 * (p / T^2) *(0.034+*dT/dz* ) + (11 / T) * *de/dz *}

The Baselga paper is a good background document that describes the different approximations and techniques surveyors have used for atmospheric refraction correction. "Practical Formulas for the Refraction Coefficient" 2014 by Baselga et. al. This paper also has an example of how significantly refraction corrections can vary with different atmospheric conditions, even at a single site over the course of 24 hours.