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. We present a novel retrieval for upper-tropospheric humidity (UTH)from High-resolution Infrared RadiationSounder (HIRS) channel 12 radiances that successfully bridges thewavelength change from 6.7 to 6.5  µ m that occurred fromHIRS/2 on National Oceanic andAtmospheric Administration satellite NOAA-14 to HIRS/3 on satellite NOAA-15.The jump in average brightness temperature (in the water vapour channel; T12 ) that this change had caused (about −7  K) could be fixed with a statistical inter-calibration method ( Shi and Bates ,  2011 ) . Unfortunately, the retrieval of UTHi(upper-tropospheric humidity with respect to ice)based on the inter-calibrated data was not satisfying at the high tail of thedistribution of UTHi. Attempts to construct a betterinter-calibration in the low T12 range (equivalent to the highUTHi range) were either not successful ( Gierens et al. ,  2018 ) orrequired additional statistically determined corrections to themeasured brightness temperatures ( Gierens and Eleftheratos ,  2017 ) . The new method presented here is based on the original one ( Soden and Bretherton ,  1993 ; Stephens et al. ,  1996 ; Jackson and Bates ,  2001 ) , but it extendslinearisations in the formulation of water vapour saturationpressure and in the temperature dependence of the Planck function tosecond order. To achieve the second-order formulation we derive theretrieval from the beginning, and we find that the most influentialingredient is the use of different optical constants for the twoinvolved channel wavelengths (6.7 and 6.5  µ m). The result ofadapting the optical constant is an almost perfect match between UTHdata measured by HIRS/2 on NOAA-14 and HIRS/3 on NOAA-15 on 1004common days of operation. The method is applied to both UTH andUTHi. For each case retrieval coefficients are derived. We present a number of test applications, e.g. on computedbrightness temperatures based on high-resolution radiosondeprofiles, on the brightness temperatures measured by the satelliteson the mentioned 1004 common days of operation. Further, we presenttime series of the occurrence frequency of high UTHi cases, and weshow the overall probability distribution of UTHi. The two latterapplications expose indications of moistening of the uppertroposphere over the last 35 years. Finally, we discuss the significance of UTH. We state that UTHalgorithms cannot be judged for their correctness or incorrectness,since there is no true UTH. Instead, UTH algorithms should fulfill anumber of usefulness postulates, which we suggest and discuss.

On the interpretation of upper-tropospheric humidity based on a second-order retrieval from infrared radiances