The standard radar horizon formula
For a radar at height h metres and a target at height t metres, the radar horizon in kilometres is approximately 4.12(√h + √t). A radar on a 30 m tower sees a fighter at 10,000 m out to about 440 km. A ship radar at 20 m sees a missile skimming the waves at 5 m out to only 28 km. Height is everything.
Atmospheric refraction
The atmosphere bends radio waves downward because density decreases with altitude. Under standard conditions this effectively increases radar range by about 15% — the '4/3 Earth' model used in planning. But conditions are rarely standard.
Ducting and super-refraction
Temperature inversions trap moisture and create a layer where refractive index decreases unusually fast. Radar signals bounce along this layer like a waveguide, extending detection from 50 km to 500 km or more. The effect is strongest at UHF and L-band. Mediterranean summers and Persian Gulf mornings are famous for it. So is the radar-operator confusion it causes.
Beating the horizon
Over-the-horizon radar uses ionospheric bounce. Airborne early warning puts the radar at 10,000 m. Satellite SAR looks down from orbit. Every solution is expensive. The physics of the horizon is why the world's most powerful nations invest billions in platforms whose only purpose is to lift the sensor higher.