Any molecule emitting LWIR above 3 meters will never reach the earths surface.
Why not?
Billy is thinking of the optical depth which is a measure of how deep radiation penetrates a substance. The amount of radiation through the substance drops exponentially with depth. It is also related to the transmittance of the substance.
Billy is quite wrong with his numbers. Back radiation at 3 meters is not attenuated by 1 / 10¹⁷. It would be attenuated by roughly 1/2. However, at above 100 meters, the exponentially drop-off will make it quite small, and for practical purposes, negligible.
Here is the way I look at it.
Above a 100 meters, the CO2 part of the atmosphere is swarming with CO2 in vibration states which are at equilibrium because of the Equipartition of energy. At that altitude the CO2 obtains it's vibration energy from thermal collisions, and has long “forgotten” what was happening nearer the earth surface. At those higher altitudes 15 micron radiation is minimal because collisions are the major energy transfer mechanism. But above the stratosphere, it's a different story when collisions are rare and radiation is the only way for the earth's energy to escape.
However
near the earth surface (less than a few dozen meters) the capturing of radiation by GHGs increases the population of the vibration part of the thermal energy, which means that the energy is not equally partitioned anymore, but has an abundance of energy in the vibration mode. In the natural tendency to achieve equilibrium at lower altitudes the excess vibration energy is naturally transferred to kinetic energy of N2 and O2, and other traces gases. The atmosphere near earth thereby heats up. Vibration energy of GHGs near the surface will never be in equilibrium because of Earth's constant radiation outflux. Back radiation also occurs.
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