The vertical profile matters
Pressure, temperature and humidity affect refractive index. Light bends toward higher index. A profile that changes with height can produce several paths from one object, stretched or inverted images, and changing visibility. A single universal correction cannot describe every such layer.
Effective radius versus layered rays
For a nearly constant gradient, changing the effective Earth radius is a convenient sight-line approximation. The ray calculator instead integrates one launch through two continuous index-gradient layers. Change the boundary height to see why the same surface reading need not imply the same full path.
What to record beside a photograph
Measure observer and target height, distance, water level, time and lens settings. Record weather at several heights if possible. A surface thermometer alone does not determine the gradient along kilometres of a ray path. Repeat observations as the profile changes rather than assigning one coefficient after seeing the image.