# Towards a Fast, Open-Path Laser Hygrometer for Airborne Eddy Covariance Measurements
## 2 Materials and Methods
### dTDLAS Methodology  
The basic principle of dTDLAS has been described in detail in [27; 28]. The relationship between the intensity of the laser beam and the absorption due to water vapor inside the cell can be described by an extended version of the Lambert-Beer law [29]:  
\[\mathrm{I}(\lambda)\mathrm{=E(t)+I_{0}(\lambda)\cdot Tr(t)\cdot exp[-S(T)\cdot g (\lambda-\lambda_{0})\cdot N\cdot L]} \tag{1}\]  
Figure 1: Functional block diagram of a dTDLAS hygrometer. In addition to the terms of the standard Lambert-Beer equation for light intensity without absorption \(\mathrm{I}_{0}(\lambda)\), the line strength \(\mathrm{S}(\mathrm{T})\), the normalized line shape function \(\mathrm{g}(\lambda-\lambda_{0})\), the absorber number density \(\mathrm{N}\) and the optical pathlength \(\mathrm{L}\), the equation also accounts for stray light and background radiation \(\mathrm{E}(\mathrm{t})\) reaching the detector, as well as broadband transmission losses \(\mathrm{Tr}(\mathrm{t})\), caused, e.g., by mirror contamination. The water vapor amount fraction \(\mathrm{c}\) can be calculated by combining Equation (1) with the definition of the concentration in amount fraction and the ideal gas law to  
\[\mathrm{c}=-\frac{\mathrm{k}_{\mathrm{B}}\cdot\mathrm{T}}{\mathrm{S}(\mathrm{ T})\cdot\mathrm{L}\cdot\mathrm{p}}\int\ln\left(\frac{\mathrm{I}(\mathrm{v})- \mathrm{E}(\mathrm{\tau})}{\mathrm{I}_{0}(\mathrm{v})\cdot\mathrm{Tr}(\mathrm{ \tau})}\right)\frac{\mathrm{d}\mathrm{v}}{\mathrm{dt}}\;\mathrm{dt} \tag{2}\]  
The individual contributions to the amount fraction \(\mathrm{c}\) are directly measured quantities, such as the temperature \(\mathrm{T}\), the pressure \(\mathrm{p}\) and the absorption signal \(\mathrm{I}(\mathrm{v})\). Constant parameters, such as the Boltzmann constant \(\mathrm{k}_{\mathrm{B}}\), and characteristic parameters of the instrument, such as the optical pathlength \(\mathrm{L}\) and the dynamic tuning behavior of the laser \(\frac{\mathrm{d}\mathrm{v}}{\mathrm{dt}}\), are also used. The spectral line strength \(\mathrm{S}(\mathrm{T})\) is taken from the HITRAN database, as well as the measured data from previous publications [25; 30; 31; 32; 33]. The emission offset \(\mathrm{E}(\mathrm{\tau})\) is measured at the beginning of each scan when the laser current is below the laser threshold so that the laser output is zero, rendering it possible to determine the zero line. The remaining parameters can be fitted using a Voigt profile for the line shape and a polynomial term for the baseline \(\mathrm{I}_{0}(\mathrm{v})\), i.e., the light intensity at the detector without absorption. Because the quantities \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{L}\), \(\mathrm{S}(\mathrm{T})\) and \(\frac{\mathrm{d}\mathrm{v}}{\mathrm{dt}}\) are all traceable to SI units, the use of the complete physical model given in Equation (2) establishes a direct link between the \(\mathrm{H}_{2}\mathrm{O}\) concentration and the SI, without the need to calibrate the dTDLAS hygrometer with, e.g., a water vapor generator. The used spectral line at 1369 nm (7299.43 cm\({}^{-1}\)) was selected because of (A) minimal cross-sensitivities to other species and a minimized amount of neighboring \(\mathrm{H}_{2}\mathrm{O}\) lines; (B) the compatibility with cost-effective, high-quality telecommunication lasers and detectors; and (C) since accurate and validated \(\mathrm{H}_{2}\mathrm{O}\) line data are available [25; 30; 31; 32]. While the cross-sensitivities to other species are minimal, there are still 18 weaker water spectral lines influencing the main line at 7299.43 cm\({}^{-1}\). To keep the degrees of freedom of the fit low, which improves the fit stability, most of the parameters of the 19 spectral lines are precalculated using spectral line data as well as measured temperature and pressure. Only the line area, the Lorentz full width at half maximum (FWHM) of the Voigt profile and the line position of the main line, as well as the three coefficients of the baseline polynomial, are left as free-fitting parameters.