Atmospheric Refraction
  Calculations of atmospheric refraction are generally based on a simplified model of atmospheric density in the troposphere which assumes that the temperature decreases at a constant lapse rate from sea level up to a height equal to eleven km, and that afterwards it remains constant. In this model, the temperature divided by the lapse rate determines the length scale in the calculations for altitudes less than this height. But daily balloon measurements across the U.S.A. reveal that in some cases the air temperature actually increases from sea level up to a height of about one km, and only after reaching a plateau, it decreases at an approximately constant lapse rate. Moreover, in three examples considered here, the temperature does not remain constant at eleven km , but continues to decreases to a minimum at about sixteen kilometers , and then increases at higher altitudes at a lower rate. Calculations of atmospheric refraction based on this atmospheric data is compared with the results of simplified models. 
Michael Nauenberg
physics.ao-ph astro-ph.EP