## Appendix: Details of SNR and Error Calculations for the IPDALLS System  
This appendix details the equations and assumptions used to assess the expected performance of the IPDALLS system. A simplified power budget is used to estimate the returned signal power and SNR for the limb sounding measurements:  
\[P_{r}=P_{0}-(P_{G}+P_{A}+P_{S}+P_{\eta}+P_{D}). \tag{1}\]  
Average power loss, defined as the difference between optical power sent out by the transmitter (\(P_{0}\)) [W] and returned to the receiver (\(P_{r}\)) [W], is mainly caused by geometric beam broadening (\(P_{G}\)), molecular absorption (\(P_{A}\)), molecular and aerosol scattering (\(P_{S}\)), losses due to retroreflector and receiver efficiency (\(P_{\eta}\)), and spectral divergence, refractive dilution (defocusing loss) and refractive irregularities (\(P_{D}\)). Spectral divergence from differential ray refraction and other limb sounding effects, including scintillation, have not been calculated for this preliminary performance analysis, although they may be significant. All power losses and noise sources except for molecular absorption and defocusing are assumed to be independent of the sounding wavelength. Geometric beam broadening is expected to dominate the power loss of the laser link or active measurement at all wavelengths, except in the denser atmosphere where (molecular) absorption and (to a lesser extent) scattering will prevail. Beam broadening is estimated by adapting the formula utilized by [PERSON] _et al._[55], which reflects standard approaches used in laser ranging. We estimate the returned signal power at the receiver, \(P_{r}\), as  
\[P_{r}=\frac{16}{\pi^{2}}\frac{P_{0}}{\Theta_{l}^{2}\Theta_{r}^{2}R^{4}}T_{\text {fwd}}T_{\text{bwd}}\phi_{\text{fwd}}\phi_{\text{bwd}}A_{\text{ref}}A_{\text {rec}}\eta_{\text{ref}}\eta_{\text{sys}}, \tag{2}\]  
where \(\Theta_{l}\) [rad] is the full divergence of the emitted laser beam, \(\Theta_{r}\) [rad] the full divergence of the reflected beam, \(R\) [m] the distance (range) between the primary spacecraft and the retroreflector spacecraft, \(A_{\text{ref}}\) [m\({}^{2}\)] the reflector area, \(A_{\text{rec}}\) [m\({}^{2}\)] the receiving telescope aperture, \(\eta_{\text{ref}}\) [-] the reflector efficiency and \(\eta_{\text{sys}}\) [-] the overall system optical efficiency. Losses due to absorption and scattering are given by the total transmission \(T_{\text{path}}\) [-], expressed as \(T_{\text{abs}}T_{\text{sca}}\), and will differ on the forward and backward paths due to induced Doppler shifts. Losses due to defocusing are taken into account as forward and backward refractive dilution factors \(\phi_{\text{fwd,bwd}}\) [-], which for simplicity and as a first order approximation have been derived following [PERSON] _et al._[54], with the same refractive index profile as has been used for estimating the solar background (see below) and atmospheric scale heights defined by the standard temperature profile. Herein, the narrow laser and reflected beam divergences are factored in as a sort of transmitter and receiver gain from geometrical considerations, and \((\eta_{\text{ref}}A_{\text{ref}})\) is the equivalent of the target surface cross section in the classical radar equation.