## Appendix: Details of SNR and Error Calculations for the IPDALLS System
Conceptually, the limb column-averaged water vapour mixing ratio can be inferred from the measurement of the forward and backward differential optical depths extracted from full pulse return signals at two wavelengths, as given here for the two-way transmission:  
\[T_{\text{fwd,(on,off)}}T_{\text{bwd,(on,off)}}=\exp\big{(}-\big{(}\big{(} \sigma^{A}_{\text{(on,off)}}+\sigma^{A}_{\text{(on+\Delta,off+\Delta)}}\big{)} \,N+\beta_{\text{(on,off)}}+\beta_{\text{(on+\Delta,off+\Delta)}}\big{)}\,R \big{)}. \tag{3}\]  
The online wavelength (subscript 'on') is centred on a water vapour absorption feature and the offline wavelength (subscript 'off\") resides in the wing, but close enough to the online frequency for attenuation by other atmospheric constituents to be essentially the same, whilst \(\Delta\) refers to the additional Doppler red-shift experienced on the backward path with respect to the forward one. The expression within the exponential corresponds to twice the optical depth, where \(\sigma^{A}\) [m\({}^{2}\)] is the water vapour absorption cross section, \(N\) [m\({}^{-3}\)] is the vapour's path-averaged concentration or number density, and \(\beta\) [m\({}^{-1}\)] is the total extinction due to absorption and scattering by other atmospheric constituents. It is then in theory straightforward to extract the average concentration from the logarithm of the ratio between the two signals, provided the absorption cross sections are known and the power ratio of the outgoing pulses is measured. If we assume that differential spectral refraction in the atmosphere and the effect of the time-delay between online and offline pulses are negligible, i.e., that the optical path covered by both wavelengths, as well as their Doppler shifted equivalents, is essentially the same, we can get a simplified expression for the two-way path-averaged differential optical depth for water vapour \(\tau_{wv}\), which will still need to be deconvolved to take into account the limb sounding geometry:  
\[\tau_{wv} \equiv N\left(\sigma^{A}_{\text{on}}R_{\text{on}}+\sigma^{A}_{ \text{on+\Delta}}R_{\text{on+\Delta}}-\sigma^{A}_{\text{off}}R_{\text{off}}- \sigma^{A}_{\text{off+\Delta}}R_{\text{off+\Delta}}\right)\] \[\cong N\left(\sigma^{A}_{\text{on}}+\sigma^{A}_{\text{on+\Delta} }-\sigma^{A}_{\text{off}}-\sigma^{A}_{\text{off+\Delta}}\right)R\] \[=\ln\frac{P_{r,\text{off}}}{P_{r,\text{on}}}+\left[\ln\frac{P_{0, \text{on}}}{P_{0,\text{off}}}+2\ln\frac{\Theta_{I,\text{off}}}{\Theta_{I,\text {on}}}+2\ln\frac{\Theta_{r,\text{off}}}{\Theta_{r,\text{on}}}+\ln\frac{\eta_{ \text{ref, on}}}{\eta_{\text{ref, off}}}+\ln\frac{\eta_{\text{sys, on}}}{\eta_{\text{sys, off}}}\right.\] \[\qquad\qquad-2\left(\beta_{\text{on}}+\beta_{\text{on+\Delta}}- \beta_{\text{off}}-\beta_{\text{off+\Delta}}\right)\right]. \tag{4}\]  
Power loss due to absorption (\(P_{A}\)) at the four water vapour IPDALLS wavelengths and in limb sounding geometry has been computed as atmospheric transmission (\(T_{\text{abs}}\)) values using the MIPAS Reference Forward Model (RFM v4.28) radiative transfer algorithm [76] in conjunction with the HITRAN 2008 database [53] and the FASCODE Model 6 (US Standard Atmosphere plus minor constituents 19 DEC99) [77]. Power loss through attenuation by Rayleigh and Mie scattering (\(P_{S}\)) is less strongly wavelength dependent than absorption. In a first approximation, we have ignored aerosol Mie scattering for the estimation of transmission losses at the sounding altitudes of interest. Atmospheric transmission (\(T_{\text{sca}}\)) after molecular Rayleigh scattering has been computed for the water vapour offline and the transmitter harmonic wavelengths in the 1976 US Standard Atmosphere.