Carbon dioxide saturation is the fact that most of the infrared radiation emitted at the surface that will be absorbed by the atmosphere is done in the lowest atmospheric layers. Computations show that one third of the absorption occurs within 25 feet of the surface, 80 % occurs within 800 feet and 99% within the first mile of the surface. Any radiation that is emitted in the wavelength of absorption by CO2 has to be reradiated at higher levels. Thus the amount emitted depends on the carbon dioxide in the upper atmosphere and its temperature.
With increased CO2, the radiation at the surface will be absorbed closer to the surface. With the atmospheric reradiation, there will be a small increase during the day and a slight decrease, due to temperature inversion conditions, at night. This is the opposite of what is observed in the past 50 years with nighttime temperatures rising more than the day time temperature.
One question that is widely known is that the climate models are predicting temperatures higher than observed. I have not seen any mathematical analyses that are concerned with the large amount of radiation observed by the carbon dioxide within 1000 feet of the surface. Most of the vertical layers in the atmospheric models have 30 to 50 layers for the atmosphere with the lowest layer being 400 to 1000 feet in thickness.