In Note 4 on the diagram, it is said that the longwave infrared radiation is emitted from the surface of the Earth due to the 168 W/m^2 of radiation from the sun that the surface has absorbed. In Note 5, it states that some of this radiation is absorbed by greenhouse gas molecules with half being emitted in the direction of space and half emitted back toward the Earth's surface. The infrared radiation from the greenhouse gas emission toward the surface is absorbed and the surface gains more heat. This additional heat causes more infrared radiation to be emitted, which again is absorbed by greenhouse gas molecules and half again is re-emitted toward the surface heating it still more.
This is very important in the greenhouse theory because the Earth's surface is desperately cold due to direct solar heating, since the black body temperature corresponding to a thermal power of 168 W/m^2 is only 233.3 K or -39.8 C. Fortunately, the theory claims half of the surface radiated infrared energy that is returned to the surface provides more heating of the surface, which radiates still more energy as it warms, but half of that is returned. In fact, the claim is made that a great deal of heat energy is added to the Earth's surface in this way. Let us examine the frequently cited Kiehl-Trenberth Energy Balance diagram to see how much.
This diagram comes in several versions and the incoming radiation from the sun absorbed by the surface differs somewhat in the various versions. In this diagram, it is 161 W/m^2, or a bit less than the 168 W/m^2 of the IPCC diagram above. The total energy added to the surface by greenhouse gas re-emission is said to be 333 W/m^2 here. This means that greenhouse gases are adding slightly more than two times the energy of the sun to warm the surface. They believe that a factor of two times the solar warming is due to the multiplier effect of greenhouse gas re-emission. Each time a greenhouse gas molecule absorbs ground radiation energy, it sends half of it back to the surface. The surface radiates all of that added energy and half of that half is returned to the surface by the gracious greenhouse gas molecule. The result is a geometric series of one-half raised to a power times the initial heat energy the surface had from the sun. It works like this:
Another interesting implication is that the absorbing greenhouse gas molecule always re-emits 100% of the infrared radiation it absorbs. It never transfers any energy to nitrogen, oxygen, or argon due to collisions, despite molecular collisions occurring at the rate of 6.9 billion per second at sea level. In reality, such collisions do occur before re-emission does and the absorbed energy often is transferred to nitrogen, oxygen, and argon molecules or atoms, which remain cooler than the Earth's surface under most conditions and times of day.