As I'd already said and referenced in response to Cabse's question, the fact that the colder upper atmospheric layers have much less water vapour was known at the end of the 19th century, establishing the theoretical basis for a greenhouse contribution by other gases even in absorption bands which are saturated in the lower atmosphere; a possibility which was confirmed when computerised layer by layer calculations became available. CO2 was obviously the first gas of interest in that regard:
https://www.aip.org/history/climate/co2.htm
Still more persuasive was the fact that water vapor, which is far more abundant in the air than carbon dioxide, also intercepts infrared radiation. In the crude spectrographs of the time, the smeared-out bands of the two gases entirely overlapped one another. More CO2 could not affect radiation in bands of the spectrum that water vapor, as well as CO2 itself, were already blocking entirely.(8) . . . .
After Ångström published his conclusions in 1900, the small group of scientists who had taken an interest in the matter concluded that Arrhenius's hypothesis had been proven wrong and turned to other problems. Arrhenius responded with a long paper, criticizing Koch's measurement in scathing terms. He also developed complicated arguments to explain that absorption of radiation in the upper layers was important, water vapor was not important in those very dry layers, and anyway the bands of the spectrum where water vapor was absorbed did not entirely overlap the CO2 absorption bands. . . .
So even if water vapor in the lower layers of the atmosphere did entirely block any radiation that could have been absorbed by CO2, that would not keep the gas from making a difference in the rarified and frigid upper layers. Those layers held very little water vapor anyway. And scientists were coming to see that you couldn't just calculate absorption for radiation passing through the atmosphere as a whole, you had to understand what happened in each layer — which was far harder to calculate.
Digital computers were now at hand for such calculations. The theoretical physicist Lewis D. Kaplan decided it was worth taking some time away from what seemed like more important matters to grind through extensive numerical computations. In 1952, he showed that in the upper atmosphere, adding more CO2 must change the balance of radiation.(25)
But would adding carbon dioxide in the upper layers of the air significantly change the surface temperature? Only detailed computations, point by point across the infrared spectrum and layer by layer down through the atmosphere, could answer that question. By 1956, such computations could be carried out thanks to the increasing power of digital computers. The physicist Gilbert N. Plass took up the challenge of calculating the transmission of radiation through the atmosphere (he too did it out of sheer curiosity, as a diversion from his regular work making calculations for weapon engineers). He nailed down the likelihood that adding more CO2 would increase the interference with infrared radiation.
The radiative forcing from CH4 changes between 1750 and 2011 is estimated at 0.48W/m^2 (+/- 0.05; IPCC AR5 WG1 Section 8.3.2.2).
IPCC AR5 WG1 Figure 8.15