Hello everybody,

thought you might be interested in some science concerning measurements of distances and heights in the lower atmosphere which has been applied in the real world for decades now. References given at the end of this post span the time from 1979 to 2016. Hyperlinks to the original articles are provided.

Bedford Level Experiment

https://wiki.tfes.org/Bedford_Level_Experiment

It is well-known that the measurements in surveying over distances of some miles can be affected by atmospheric refraction due to the change in index of refraction of air with height. Such changes are due to change in temperature and relative humidity with height above ground. I do not conѕider the latter in this post.

References [1], [2] and [3] (at the bottom of this post) present this phenomena in terms of a coefficient of refraction, k, defined as :

k = R / r

where R is the radius of earth and r the radius of the circle describing the path of the refracted light beam. Hence k=0 represents the case when the light is not refracted and follows a straight line from the target to be investigated to the observer. For k=1 the light would follow exactly the curvature of a round earth of radius R = 6370 km. Negative values for k correspond to a case with the light path curving upwards.

In the above mentioned references a mathematical expression for the value of k is presented :

k = 503*p*(0.0343 + dT/dh)/(T*T)

p = atmospheric pressure in mbar (1015 mb = 14.7 psi)

T = absolute temperature in degree Kelvin (288 K = 59 Fahrenheit)

dT/dh = change of temperature with height in Kelvin/meter.

The values for p and T are mine representing an average atmospheric condition.

If the temperature does not change with height then dT/dh = 0 and the value of k becomes k=0.21 with the above numbers for pressure and temperature.

Close the ground dT/dh can be expected to be positive if the ground is cooler than the air above. Higher up we observe in general that the temperature decreases with height and therefore a negative dT/dh is common in that region.

In order to get a feeling for what numbers we are talking about, let's assume the air temperature changes by +0.13 deg kelvin ( = 0.234 Fahrenheit) per vertical meter ( about a yard ). We get k=1.01. In that case, on a round earth, any object a few miles away would still appear to sit on the horizon in full view regardless of distance. An observer who is not familiar with refraction will therefore conclude that the earth is flat.

Reference [2] cites a variety of typical ranges for k depending on at which height above ground geodesic measurements are taken.

In the region of 100 m (330 ft) and above the temperature gradient is fairly constant at dT/dh = −0.006 K/m resulting in values of k around 0.17 .

For heights between 20-30 m up to a 100m dT/dh = −0.01 K/m resulting in k=0.15. For these values of k the bending of light due to refraction would be still small in comparison to the curvature of an earth with a radius of 6370 km and line-of-sight measurements would give pretty conclusive evidence about the flatness of earth's surface. But, both, target and observer and anything in between has to be more than 20m ( 65 ft ) above ground. Even if that is the case, a really good experiment will be accompanied by precise measurements of the temperature gradient at that height.

Below 20-30m the temperature gradient in the air is subject to the thermal properties of the surface underneath. [2] cites several experimental studies with values for the coefficient of refraction, k, ranging from -14 to +18 as distance to the ground decreases to below 10m (33 ft). Based on that and my calculations with dT/dh=0.13 K/m giving already a value of k=1.01 it is clear that line-of-sight experiments conducted close to the ground must be accompanied by very precise measurements of the temperature gradient along the path of light.

**In summary : the accuracy of the measurement of the temperature gradient must better than, let's say, a few hundreds of a degree per vertical meter in order for an experiment to prove or disprove conclusively the flatness of earth's surface.**[1] D. Gaifillia, D et.al.

"Empirical Modelling of Refraction Error in Trigonometric Heighting Using Meteorological Parameters"

Journal of Geosciences and Geomatics, Vol. 4, No. 1, 2016, pp 8-14

http://pubs.sciepub.com/jgg/4/1/2/index.html

[2] Hirt, Christian et.al.

"Monitoring of the refraction coefficient in the lower atmosphere using a controlled setup of simultaneous reciprocal vertical angle measurements"

Journal of Geophysical Research: Atmosphere, Volume 115, Issue D21; Nov 2010

https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2010JD014067[3] Fraser, C.S

"Atmospheric Refraction Compensation in terrestrial Photogrammetry"

Photogrammatic Engineering and Remote Sensing, Vol 45, No.9, September 1979, pp.1281-1288

https://www.asprs.org/wp-content/uploads/pers/1979journal/sep/1979_sep_1281-1288.pdf