How this all works, physically:

1. Incident flux from the sun on the top of our atmosphere F_o is filtered through the atmosphere by its short wavelength transmittence T_s , where 0 < T_s < 1.0

2. The flux that reaches the ground is then F_o * T_s

3. That flux is absorbed by the ground and the ground heats up to some temperature (T_g).

4. The ground then re-radiates that heat as outgoing long wavelength IR radiation (F_g).

5. Some of that flux (F_g) is absorbed by the atmosphere through its long wavelength transmittence T_t , where 0 < T_t < 1.0

6. The combination of absorption of short (s) and absorption of long (t) wavelength radiation causes the atmosphere to heat up to a temperature (T_a) and then radiate that temperature away as flux (F_a); 1/2 of it up (into space) and 1/2 of it down (back to ground)
   
   
7. Since as much flux goes out of the system as comes into the system we can set up the following equilibrium conditions.
   Top of the atmosphere: F_o = F_a + T_t *F_g

   At the Ground: F_g = F_a + T_s *F_o

8. Let F_a = F_o - T_t *F_g and substitute then; eventually get that
   F_g = F_o *(1+T_s)/(1+T_t)

   For our atmosphere: T_s = 0.9; T_t =0.2 F_g = 1.6*F_o which leads to a 30-35K increase in the nominal surface temperature.