## 2 Theory
### Differential Absorption Measurement in Transmission  
versus _Reflection_  
Differential absorption techniques for measuring atmospheric gases began appearing shortly after the advent of the laser. The literature describing these various approaches is vast and spans both ground and space concepts. A good general reference can be found in [17] and a few additional selected examples for the interested reader can be found in references [18; 19; 20; 21; 22; 23; 24]. Only a brief overview of the concepts is provided here, as needed for clarification of the LAnTeRN measurement concept. Starting with the standard lidar equation for the received signal power \(P_{rec}\) at wavelength \(\lambda_{0}\) from a volumetric scattering target of \(\Delta R\), centered at range \(R\), given by:  
\[P_{rec}(\lambda_{0},R)=\ P_{L}\frac{A_{rec}}{R^{2}}\times\eta(\lambda_{0}) \times\xi(R)\times\beta_{\pi}(\lambda_{0},R)\times\Delta R\times e^{-2\int_{0} ^{R}\kappa(\lambda_{0},r)dr} \tag{6}\]  
where \(P_{L}\) is the transmitted laser power, \(A_{rec}/R^{2}\) is the solid angle subtended by the receiver, \(\eta(\lambda_{0})\) is the optical efficiency of the receiver system at \(\lambda_{0}\), \(\xi(R)\) is a geometric factor describing the overlap of the laser and the receiver telescope as a function of range, \(\beta_{\pi}(\lambda_{0}\), \(R)\) is the atmospheric volumetric backscatter coefficient at range \(R\) for wavelength \(\lambda_{0}\), and is assumed to be homogenous over a slice of the atmosphere of thickness \(\Delta R\) centered at \(R\), and \(\kappa(\lambda_{0}\), \(r)\) is the volumetric extinction coefficient which is integrated from 0 to range \(R\) and includes both absorption and scattering into angles other than the receiver view angle, \(\theta\
eq 180^{\circ}\) for this standard lidar description. The factor of 2 in the exponent is based on a standard lidar making the measurement in reflection and accounts for the round trip attenuation. For the case of the LAnTeRN arrangement the standard lidar equation in Equation (6) can be simplified to,  
\[P_{rec}(\lambda_{0},R)=\frac{P_{L}(\lambda_{0})}{A_{Las}(R)}\times A_{rec} \times\eta(\lambda_{0})\times\mathrm{e}^{\left(-\int_{0}^{R}\kappa(\lambda_{ 0},r)dr\right)} \tag{7}\]  
where we have assumed the area of the transmitted laser signal at range \(R\) (\(A_{Las}(r)\)) overfills the receiver aperture. The received power is only dependent on the power density at the receiver, the receiver area, the receiver efficiency, and the one-way atmospheric extinction between the transmitter and receiver. By selecting two closely spaced (\(<\)100 pm) wavelengths corresponding to an online and offline wavelength, where online refers to a wavelength with significant absorption by the gas of interest and offline refers to a wavelength with much less absorption by the gas of interest, one can apply the standard differential absorption (DA)technique. \[\frac{P_{rec}(\lambda off,R)}{P_{rec}(\lambda on,R)} =\frac{P_{Loff}}{P_{Lon}}e\big{(}l_{0}^{R}\kappa(\lambda_{on}r)dr -l_{0}^{R}\kappa(\lambda_{off}r)dr\big{)} \tag{8}\] \[\cong\frac{P_{Loff}}{P_{Lon}}e\big{(}l_{0}^{R}\alpha(\lambda_{on}r )dr-l_{0}^{R}\alpha(\lambda_{off}r)dr\big{)}\]  
where, \(e\big{(}l_{0}^{R}\kappa(\lambda_{on}r)dr-l_{0}^{R}\kappa(\lambda_{off}r)dr \big{)}\) is the differential transmission between the online and offline wavelengths to range \(R\), from the Beer-Lambert law. The DA technique relies on an assumption that the wavelength dependence of the extinction term, both for scattering and absorption, from all but the molecule of interest is negligible. This is a reasonable assumption given the close spacing of the online and offline wavelengths and proper selection of the wavelengths relative to other interfering absorption features. Using this assumption we get the result on the right hand side of Equation (8) where the general extinction coefficient, \(\kappa\), is replaced by the absorption coefficient of the molecule of interest, \(\alpha\).