# MERLIN: A French-German Space Lidar Mission Dedicated to Atmospheric Methane
## 3 Methodology
The integration runs from top of the atmosphere (TOA) down to the target distance \(r_{T}\). On the right-hand side of Equation (2) a further term appears that sums all other DAOD contributions from interfering gases, notably H\({}_{2}\)O and CO\({}_{2}\). In principle, the contribution of these spectroscopic terms can be minimized by a careful selection of the quasi-monochromatic laser frequency position within a chosen CH\({}_{4}\) absorption band, which should be almost free from interfering gases. As a unique side effect of our approach, investigations revealed that the water vapour interference can be significantly further reduced by applying the so-called water vapor compensation mode [55; 88]. It is based on the idea of choosing the offline position within a water vapour line such that a measurable water vapor DAOD\({}_{\text{H2O}}\) appears, which then partly compensates the water vapour contribution related to the dry-air mixing ratio calculation (left hand side of Equation (2)). It is worth mentioning that the scattering and extinction properties of the atmosphere can be regarded to be identical for both frequencies and will not contribute to a measurement bias, which is a key asset of the proposed measurement principle. A further advantage of using pulsed lasers relates to the known light path, which is indisputably defined by the viewing geometry of the Lidar instrument and the round-trip time of the transmitted laser pulses with typical pulse lengths of a few tens of nanoseconds. However, small amounts of multiple scattering from penetration of optically thick aerosol layers and optical thin cirrus clouds could impact on the length of the light path, and thus can give rise to a measurement error. This impact will be suppressed by choosing a narrow transmitter divergence in conjunction with a narrow receiver field of view. For modelling purposes it is convenient to introduce pressure coordinates using the hydrostatic equation in combination with the ideal gas law. By this convention, the column-integrated dry-air mixing ratio of CH\({}_{4}\) can be calculated from measured DAOD using the following identity:  
\[XCH_{4}\equiv\frac{\int_{0}^{p_{T}}m_{r}(p)WF(p)dP}{\int_{0}^{p_{T}}WF(p)dp}= \frac{DAOD}{\int_{0}^{p_{T}}WF(p,T)dp}=\frac{\ln\left[\frac{p_{off}(r_{T})E_{ opt}}{P_{on}(r_{T})E_{off}}\right]-DAOD_{other~{}gases}}{2\int_{0}^{p_{T}}WF(p,T)dp} \tag{3}\]  
where  
\[WF(p)=\frac{\sigma_{on}(p,T)-\sigma_{off}(p,T)}{g(p)M_{air}(1+M_{H_{2}O}q(p)_{ dry})}\]  
denotes the weighting function. This function contains the differential absorption cross section for methane, given as function of pressure and temperature, \(g\) is the acceleration of gravity, \(M_{air}\) and \(M_{H2O}\) are the masses of dry air and water vapour molecules, and \(q_{dry}\) is the water vapour mixing ratio with respect to dry air. Information of the observed air mass is obtained from the integration of \(WF\), spanning the column between pressure 0 at TOA and \(p_{T}\) the pressure at the location of the target (e.g., laser footprint on the target). Note that to obtain this information by passive remote sensing, an additional oxygen channel is required. If the target is the Earth's surface, \(p_{T}\) will be the surface pressure \(p_{surf}\). It will be derived using operational analyzes from Numerical Weather Prediction (NWP) centres, after correction for a pressure change given by the target height, commonly denoted Surface Scattering Elevation (SSE), as shown in the conceptual drawing of Figure 2.  
The elevation correction can be calculated assuming hydrostatic equilibrium:  
\[p_{surf}=p_{NWP}\exp\left[\frac{SSE-Z_{NWP}}{-R.T_{NWP}(p)}g\right] \tag{4}\]  
where \(T_{NWP}(p)\), \(p_{NWP}\), and \(Z_{NWP}\) are the modelled vertical temperature profile, surface pressure and the surface elevation of the \(NWP\) analyses for the grid box in which the measurement is located. \(R\) denotes the gas constant for air.