## Appendix: Details of SNR and Error Calculations for the IPDALLS System
Solar spectral radiant flux density at the tangent point \(I_{\text{sun}}\) [Wm\({}^{2}\)nm\({}^{-1}\)] was assumed to be independent of optical depth, and the effective scattering path length \(l_{\text{eff}}\) was set to 300 km. The Rayleigh-scattered solar background power \(P_{\text{bkgrd}}\) collected by the receiver was then estimated as  
\[P_{r}=\frac{16}{\pi^{2}}\frac{P_{0}}{\Theta_{l}^{2}\Theta_{r}^{2}R^{4}}T_{\text {fwd}}T_{\text{bwd}}\phi_{\text{fwd}}\phi_{\text{bwd}}A_{\text{ref}}A_{\text {rec}}\eta_{\text{ref}}\eta_{\text{sys}}, \tag{2}\]0  
with nominal channel optical filter bandwidths \(B_{W,\text{optical}}\) of 1 nm and where FOV\({}_{\text{rec}}\) [mrad] describes the receiver full field-of-view. Noise originating within the detector is referred to as internal noise (Equation (7), second term in square brackets), and depends on the detector type. For enhanced sensitivity in the near IR and high internal gain, APDs are preferred over the traditional photomultiplier tubes (PMTs) and can be operated in either their normal linear mode or in photon-counting mode. APDs furthermore have fast rise and fall times and are recommended for very high amplifier bandwidth \(B_{W}\) applications. In order to adequately sample the narrow return pulse without excessive smoothing at low bandwidths, we specified the bandwidth through the relationship \(B_{W}\cong k/\tau_{L}\), with \(\tau_{L}\) [s] quantifying the emitted laser pulse duration and with a \(k\) factor roughly between 3 and 5. For our estimation of the CNR, we have used typical APD specifications provided by Excelitas/PerkinElmer, corresponding to models that have been employed in the GLAS laser system [82] and the WALES airborne campaign [42], notably the C30954E-DTC unit thermoelectrically cooled to \(-20\)\({}^{\circ}\)C with a double-stage cooler to reduce the dark current. The excess noise factor, related to internal amplification statistics, has been estimated following [83] with \(F=k_{\text{eff}}M+\left(1-k_{\text{eff}}\right)\left(2-1/M\right)\), where \(k_{\text{eff}}\) denotes the effective carrier ionization ratio specified as \(k=0.02\) for a Si high performance reach-through structure [84]. Surface and bulk dark currents, measured at room temperature \(T_{\text{meas}}\) [K], have been extrapolated to their values in a cooled setup (\(T_{\text{oper}}\) [K]) following the Arrhenius equation,  
\[P_{r}=\frac{16}{\pi^{2}}\frac{P_{0}}{\Theta_{l}^{2}\Theta_{r}^{2}R^{4}}T_{\text {fwd}}T_{\text{bwd}}\phi_{\text{fwd}}\phi_{\text{bwd}}A_{\text{ref}}A_{\text {rec}}\eta_{\text{ref}}\eta_{\text{sys}}, \tag{2}\]1  
where \(k_{B}\) is the Boltzmann constant and where the activation energy \(E_{a}\) [J] was set to 0.70 and 0.55 eV for the surface and bulk dark currents, respectively. A simplified expression for the modelled random error of the two-way path-averaged water vapour optical depth can be derived from the first term on the left in Equation (4), disregarding the additional differential terms which are based on unknown system parameters. Using standard first order error propagation and setting SNR\({}_{\text{(on,off)}}\equiv\sqrt{\langle P_{r,\text{(on,off)}} \rangle^{2}/\langle\Delta P_{r,\text{(on,off)}}^{2}\rangle}\)[33], we estimate the variance of the two-way optical depth and then the relative random error [%] as  
\[P_{r}=\frac{16}{\pi^{2}}\frac{P_{0}}{\Theta_{l}^{2}\Theta_{r}^{2}R^{4}}T_{\text {fwd}}T_{\text{bwd}}\phi_{\text{fwd}}\phi_{\text{bwd}}A_{\text{ref}}A_{\text {rec}}\eta_{\text{ref}}\eta_{\text{sys}}, \tag{2}\]2  
where \(n_{\text{shots}}\) gives the number of independent shot pairs. (c) 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ([[http://creativecommons.org/licenses/by/3.0/](http://creativecommons.org/licenses/by/3.0/)]([http://creativecommons.org/licenses/by/3.0/](http://creativecommons.org/licenses/by/3.0/)).)