The vertical refraction in the atmosphere cannot be ignored for longrange observations. The exact value is difficult to predict as it requires a large number of temperature, pressure and humidity measurements all along the path of observation. However, the value can be measured using reciprocal zenith angles and other methods. This has been done and quantified many times at different locations under varying conditions.
 Monitoring of the refraction coefficient in the lower atmosphere using a controlled setup of simultaneous reciprocal vertical angle measurements
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An essential finding of past research efforts is that the refraction coefficient k and vertical temperature gradient are directly related to each other [e.g., Brocks, 1939]. The refraction coefficient of a particular point, in the literature commonly referred to as the local refraction coefficient and denoted with c, is connected to the vertical temperature gradient ∂T/∂z (K/m) using [e.g., Bahnert, 1987; Joeckel et al., 2008]:where p is the pressure (HPa) and T is the temperature (K). The local refraction coefficient c is essentially a function of the temperature gradient ∂T/∂z, and depends only slightly on pressure p and temperature T [Wunderlich, 1985]. Temperature gradients allow modeling and reducing vertical refraction influences in surveying [e.g., Brocks, 1939; Heer and Niemeier, 1985; Kharaghani, 1987; Hennes, 2002; Ingensand, 2008].
 Has an overview of past studies into the effects of refraction in the atmosphere.
 “An essential finding of past research efforts is that the refraction coefficient k and vertical temperature gradient are directly related to each other [e.g., Brocks, 1939].”
 “…the thermal characteristics of the air strata of the lower atmosphere (lowest 20–30 m) are strongly subjected to the varying thermal properties of the surface [e.g., Angus‐ Leppan, 1969].”
 “In the evening, the Earth’s surface normally cools off faster than the overlaying air strata. Usually, this results in strong positive gradients ∂T/∂z [Angus‐Leppan, 1969]. Cloud cover attenuates the heat transfer from the Sun and, hence, from the ground, thus leading to smaller absolute values of the temperature gradients and refraction effects in the course of a day.”
 “Brocks[1950b] demonstrated that refraction effects in the lower atmosphere generally multiply with decreasing height. This is because heat transition from the Earth’s surface is the stronger, the less distant the atmospheric layers are. Therefore, the largest absolute values of temperature gradients are to be expected immediately above the ground.”
 “Stober [1995] determined refraction coefficients between 0 and +2 from vertical angle measurements for ground clearances between 0.5 and 4 m over an ice field in Greenland.”
 “Kabashi[2003] obtained refraction coefficients from +1 to +18 from reciprocal vertical angles for a line of sight about 5 m over water.”
 Summary of Refraction Behavior
 “…refraction continuously increases to the positive range with peaks of about +12 to +16 equally present in the data around sunset.”
 “The refraction coefficients on cloudy days generally exhibit smaller variations.”
 Conclusions:
 “Our refraction experiments, carried out across homogeneously vegetated grassland with a ground clearance of about 1.8 m (i.e., an often used working height in surveying), showed a range of k between −4 and +16 for sunny summer days.”
 “…our empirical results show that the refraction coefficient k may reach magnitudes as large as +12 to +16 over grassland at 1.8 m.”
 Publication: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2010JD014067
 PDF: https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2010JD014067
 Direct PDF Download
 Account of the Observations and Calculations, of the Principal Triangulation; and of the Figure, Dimensions and Mean Specific Gravity of the Earth as Derived Therefrom
 Alexander Ross Clarke, 1833
 Pages 542550 as marked on the pages, or 571579 of the PDF.
 “…the coefficient required is simply the ratio of the excess of the observed above the true zenith distance to the angle subtended by the two stations at the centre of the earth.”
 This means the observation appears above its true location.
 Coefficient of Refraction:
 +0.0809 for rays crossing the sea
 +0.0750 for rays not crossing the sea
 Note that over sea observations are raised more than not over the sea.
 https://books.google.com/books?id=DKELAQAAIAAJ
 Direct PDF Download
 Empirical Modelling of Refraction Error in Trigonometric Heighting Using Meteorological Parameters
 D. Gaifillia, V. Pagounis, M. Tsakiri, V. Zacharis
 Measurements over land in Greece.
 Page 3 lists results of measuring the coefficient of refraction, k, between +0.050 and +0.200
 http://pubs.sciepub.com/jgg/4/1/2/index.html
 Direct PDF Download
 Gleichzeitiggegenseitige Zenitwinkelmessung über größere Entfernungen mit automatischen Zielsystemen

Translated: “Simultaneous reciprocal Zenith Angle Measurement over Larger Distances with Automatic Target Systems“
 Ismail Kabashi 2004
 Measurements of the effects of refraction about 5 meters above the water at different times of the day.
 The coefficient of refraction was found to be between 0.5 and 17 on a clear day with a steeply increasing coefficient of refraction after sunset and peaking several hours after sunset.
 On a cloudy day, the coefficient of refraction was lower and less variable.
 https://publik.tuwien.ac.at/files/PubDat_119937.pdf
 Direct PDF Download

 “Refraktionsbestimmung auf dem grönländischen Inlandeis”
 Translated: “Refraction on the Greenland ice sheet”
 Prof. Dr.Ing. Manfred Stober
 Translated: Shown are very different temperature gradients depending on solar radiation, wind and cloud cover. The resulting refraction coefficient takes values up to about at k = 1.7 (an order of magnitude higher than usual in Germany).
 Direct PDF Download: STO_Intergeo2004_GroenlandRefraktion
 Translated PDF: STO_Intergeo2004_GroenlandRefraktion.de.en
Online calculator for refraction using air temp at observer and target by Andrew T Young:
https://aty.sdsu.edu/explain/atmos_refr/altitudes.html
Overview of Young’s pages:
https://aty.sdsu.edu/explain/atmos_refr/astr_refr.html
Other publications concerning refraction. I have reviewed some, but not all.
 An Accurate Method for Computing Atmospheric Refraction
 Astronomical refraction formulas for all zenith distances
 Atmospheric Refraction Predictions Based on Actual Atmospheric Pressure and Temperature Data
 Atmospheric Refraction. Part I
 Atmospheric Refraction. Part II
 Atmospheric Refractive Electromagnetic Wave Bending and Propagation Delay
 Errors in precise leveling
 INVESTIGATION OF REFRACTION IN THE LOW ATMOSPHERE
 Practical Formulas for the Refraction Coefficient
 PROPAGATION OF REFRACTION ERRORS IN TRIGONOMETRIC HEIGHT TRAVERSING AND GEODETIC LEVELLING
 Refraction by Earth’s Atmosphere near 12 Microns
 REFRACTION NEAR THE HORIZON
 Removal of Refraction Errors in Geodetic Levelling – 1979
 Results Of Leveling Refraction Tests by NGS TR NOS92 NGS22
 Earth Curvature and Atmospheric Refraction Effects on Radar Signal Propagation