The formula for estimating radiative forcing (RF or ΔF) from a change in atmospheric CO2 concentration is usually given as:
ΔF = 𝞪·ln(C/C₀) W/m²
C/C₀ is the ratio of new to old CO2 concentrations
Myhre 1998 & the IPCC (TAR & later) estimate:
𝞪 = 5.35 ±0.58 (which is 3.7 ±0.4 W/m² per doubling of CO2)
Happer 2013 (and 2015) reports calculating, based on corrected modeling of CO2 lineshapes, that that's ≈40% too high, which makes:
𝞪 ≈ 3.8 ±0.5 (which is 2.6 ±0.5 W/m² per doubling)
Feldman et al 2015 measured downwelling longwave IR “back radiation” from CO2,
at ground level, under clear sky conditions, for a decade. They reported that a 22 ppmv (+5.953%) increase in atmospheric CO2 level resulted in a 0.2 ±0.06 W/m² increase in downwelling LW IR
from CO2, which is +2.40 ±0.72 W/m² per doubling of CO2.
However, ≈22.6% of incoming solar radiation is reflected back into space, without either reaching the surface or being absorbed in the atmosphere. So, adjusting for having measured at the surface, rather than TOA, gives ≈1.29 × (2.40 ±0.72) per doubling at TOA, and dividing by ln(2), yields:
𝞪 ≈ 4.47 ±1.34 (which is 3.10 ±0.93 W/m² per doubling)
That's close to Halpern's “4.35”, and closer to Happer's “3.8” than to Myhre's “5.35,” but the uncertainty interval is wide enough to encompass all three estimates. It does preclude the SAR's “6.3” figure.
Rentsch 2020 (draft), analyzed AIRS satellite spectroscopy, and found that under nighttime, cloud-clear conditions, a 37 ppmv CO2 increase caused +0.358 ±0.067 W/m² radiative forcing increase at TOA, which is:
𝞪 = 3.79 ±0.71 (which is 2.62 ±0.49 W/m² per doubling)
That's about 70% of Myhre 1998, and very close to Happer's result.
UPDATE: There's a new paper on this topic:
Kramer, RJ et al. 2021. Observational evidence of increasing global radiative forcing. Geophysical Research Letters, 48, e2020GL091585. doi:10.1029/2020GL091585. (See discussion.)